pounce_algorithm/line_search/

backtracking.rs

//! Backtracking line-search driver — port of
//! `Algorithm/IpBacktrackingLineSearch.{hpp,cpp}`.
//!
//! Owns the alpha-reduction loop, max-soc / second-order-correction
//! slot, watchdog mechanism, and the fallback to restoration. Phase 7
//! ships the alpha-loop for the filter line search; SOC and watchdog
//! land alongside the restoration phase (Phase 9).
//!
//! The contract with the acceptor is the trio
//! `(theta, phi, d_phi)` at the current iterate plus the trial
//! `(theta_trial, phi_trial)` per backtracking step. Trial-point
//! construction is `x_trial = x + α·dx`, `s_trial = s + α·ds`; the dual
//! step uses the same α for the filter acceptor (upstream
//! `IpBacktrackingLineSearch.cpp:702-728` — primal-dual share α
//! when no fraction-to-the-boundary truncation differs).
//!
//! `find_acceptable_trial_point` returns `Outcome::Accepted` on a
//! successful trial, `Outcome::TinyStep` when α drops below
//! `alpha_min`, and `Outcome::Failed` when the alpha loop exhausts
//! without acceptance (which the main loop maps to a restoration
//! attempt).

use crate::ipopt_cq::IpoptCqHandle;
use crate::ipopt_data::IpoptDataHandle;
use crate::ipopt_nlp::IpoptNlp;
use crate::iterates_vector::IteratesVector;
use crate::kkt::pd_search_dir_calc::PdSearchDirCalc;
use crate::line_search::filter_acceptor::AcceptDecision;
use crate::line_search::ls_acceptor::BacktrackingLsAcceptor;
use pounce_common::types::Number;
use std::cell::RefCell;
use std::rc::Rc;

/// Number of trial points the plain geometric sequence gets to itself
/// before [`BacktrackingLineSearch::next_alpha`]'s quadratic
/// interpolation is allowed to pick the next `alpha` (gh #818).
///
/// **The interpolation is a treatment for a long line search, and it is
/// only harmless where the line search is long.** gh #818 is a
/// quasi-Newton model whose *scale* is wrong by orders of magnitude in
/// the directions its curvature pairs do not span: the acceptable step
/// is `alpha ~ 4e-6`, and halving walks there in 19-20 trial points,
/// every one of them a full objective evaluation, every iteration. A
/// line search that accepts in two or three trials does not have that
/// problem, and interpolating into it replaces a step length the filter
/// was about to accept with a different one — a trajectory change
/// bought for nothing.
///
/// Measured, `scripts/sweep-fixtures.sh` against `a5e0a837`, 156
/// fixture-legs, every row taken against that one baseline. The `exact`
/// leg is byte-identical at every value, because `alpha_red_factor_min`
/// resolves to `alpha_red_factor` there, so every line below is an
/// `lbfgs` leg. **Bold is a status the baseline did not have.**
///
/// | this constant | legs moved | `cresc4` `RestoFailed`/1323 | `deb7` `ErrInStep`/1242 | `eigena2` `ErrInStep`/252 | `square_flowsheet_resto` `Infeasible`/3000 |
/// |---|---|---|---|---|---|
/// | 0 (interpolate always) | 12 | `Succeeded`/241 | **`RestoFailed`**/2381 | 131 | **`Succeeded`**/2393 |
/// | 2 | 7 | `RestoFailed`/215 | **`MaxIter`**/3000 | 170 | `Infeasible`/49 |
/// | 3 | 6 | `Succeeded`/226 | `ErrInStep`/1327 | 109 | `Infeasible`/2022 |
/// | 5 | 4 | `Succeeded`/264 | **`RestoFailed`**/455 | 91 | unmoved |
/// | **6** (shipped) | **4** | `Succeeded`/281 | `ErrInStep`/1010 | 201 | unmoved |
///
/// Six is the only value measured that gains a status and loses none.
/// `cresc4` is the gain at every gate that reaches it; what separates 6
/// from its neighbours is `deb7`, which changes verdict at 0, 2 **and
/// 5** and keeps `ErrorInStepComputation` only at 3 and 6, and of those
/// two only 6 shortens it (1242 -> 1010 against 3's 1327). The fourth
/// moving line at 5 and 6 is one objective digit on `hs13_bigstart` at
/// an unchanged iteration count.
///
/// **Why not 5, which an earlier revision shipped.** Two reasons, and
/// the second is the one that forced the change. `deb7` above is the
/// first. The second is `python/tests/test_starts_racing.py`: one
/// `_rastrigin_eq` line search in that suite reaches exactly 5-6 trial
/// points, so a gate of 5 interpolates into it and reroutes the whole
/// multistart race. Halving-against-fixed `solver_evals` over that
/// suite reads 1.022 with the interpolation off, **1.320 at gate 5**,
/// and 1.019 / 1.022 / 1.022 at gates 6 / 7 / 8 — two red assertions,
/// at 2870 `user_evals` and 2169 `solver_evals` against budgets of
/// 2800 and 2006. Six restores that suite to its interpolation-off
/// numbers (1949 `solver_evals` against 1951 off) without giving up
/// gh #818.
///
/// **`square_flowsheet_resto` is a pre-existing wrong verdict on the
/// baseline, not something a gate value creates.** An earlier revision
/// of this comment had it the other way round — that interpolating
/// from the first trial reports a model feasible by construction as
/// converged to a point of local infeasibility. Re-measured against
/// `a5e0a837`, the *baseline* is the `InfeasibleProblemDetected`/3000
/// line and gate 0 is what turns it into `SolveSucceeded`/2393. The
/// original measurement predates gh #817, which is in `main` now and
/// moved that fixture's `lbfgs` leg. Gates 5 and 6 do not touch the
/// line either way, so this constant is not the lever for it; it is
/// recorded here so the next reader does not inherit the inverted
/// claim.
///
/// It also fixes gh #818's own model. On the issue's quadratic at the
/// default memory `m = 6` and `limited_memory_initialization=scalar1`,
/// `before` being `alpha_red_factor_min` set equal to
/// `alpha_red_factor` — upstream's fixed sequence, bit for bit:
///
/// | | before gh #818 | **gated at 6** |
/// |---|---|---|
/// | `n = 4` (the report) | 76 | **22** |
/// | `n = 8`, `m = 10` | 74 | **61** |
///
/// The 8-variable case at the *default* memory is not on that list.
/// It is not closed by this change and is not closed by any gate:
/// see `issue_818_eight_variable_default_memory_does_not_stall` in
/// `pounce-rs/tests/issue_818_lbfgs_illconditioned_quadratic.rs`,
/// which carries the two-knob sweep showing it converging in 5 of 15
/// cells with no pattern. Tuning this constant to that cell is what
/// produced the gate of 5 and the racing regression above.
///
/// One cell of the 32-cell model sweep — `n = 8` at cond 1e12 with
/// `m = 6` — exited `Diverging_Iterates` at 352 under every gate in
/// {5, 8, 12, 20} and every `alpha_red_factor_min` measured, which is
/// what it looks like when the constant is not the variable. It was
/// not: the interpolation was reaching a *watchdog* trial excursion
/// that the divergence guard in `IpoptAlgorithm::iterate` then read as
/// unboundedness, on a point [`Self::handle_watchdog_failure`] had
/// already rejected and was holding a snapshot for. With that guard
/// fixed the cell no longer claims divergence — it reports
/// `Error_In_Step_Computation` at 521, still not a solve, but at a
/// better objective than the fixed sequence reaches in four times the
/// iterations (6.4e-11 against 2.8e-10 at `max_iter`). See
/// `pounce-rs/tests/watchdog_trial_is_not_a_divergence_verdict.rs`;
/// CHANGELOG.md carries the full grid.
///
/// Not a registered option. `alpha_red_factor_min = alpha_red_factor`
/// already turns the interpolation off entirely, which is the escape
/// hatch a caller needs; a second knob between "off" and "on" would be
/// one more number with no population behind it.
const ALPHA_INTERP_MIN_TRIALS: i32 = 6;

/// gh#1016 — the approximate-Wolfe test of Hager & Zhang (*SIAM J. Optim.*
/// 16(1), 2005, eq. 4.1), used only once a solve has shown that the
/// objective cannot resolve its own decrease (see
/// [`BacktrackingLineSearch::objective_noise_limited`]).
///
/// `phi'(alpha)` must satisfy `WOLFE_SIGMA * phi'(0) <= phi'(alpha) <=
/// (2 * WOLFE_DELTA - 1) * phi'(0)`: the slope at the trial point has fallen
/// to within `1 - SIGMA` of flat but not overshot to more than `1 - 2*DELTA`
/// of the starting slope the other way. Both are the paper's values.
const WOLFE_DELTA: Number = 0.1;
const WOLFE_SIGMA: Number = 0.9;
/// The test only fires when `|phi(alpha) - phi|` is within `WOLFE_EPS * max(1,
/// |phi|)` — the paper's `epsilon_k` (there `1e-6 * |f_k|`). The slope
/// conditions are what keeps the tolerated increase small; this only rules
/// out a step the objective itself says is a real ascent.
const WOLFE_EPS: Number = 1e-6;

/// Outcome of the backtracking line search. Mirrors the booleans
/// upstream returns through `accept_` plus the `tiny_step_flag` on
/// `IpoptData`.
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum Outcome {
    /// Trial point accepted at the recorded `alpha`.
    Accepted,
    /// `alpha` fell below `alpha_min_frac` × current α₀ ⇒ tiny step.
    /// Caller maps to `STEP_BECOMES_TINY` in upstream's exception flow.
    TinyStep,
    /// All α reductions rejected; the caller hands off to restoration.
    Failed,
    /// The shared wall/CPU-time deadline was crossed mid-search
    /// (pounce#242). The caller terminates the solve with the
    /// corresponding time-limit status, returning the current best
    /// iterate (`data.curr`, left untouched — no trial was promoted).
    Deadline,
}

/// Policy for the step length applied to the equality multipliers
/// `y_c`, `y_d`. Mirrors upstream's `alpha_for_y` option (subset of
/// the upstream enum — pounce only ports the variants that the
/// Mehrotra cascade and default code paths exercise).
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum AlphaForY {
    /// Use the primal step length (upstream default).
    Primal,
    /// Use the dual step length. Selected by the Mehrotra cascade
    /// (`alpha_for_y=bound-mult`).
    BoundMult,
    /// Always take a full step on the equality multipliers.
    Full,
    /// Use the minimum of the primal and dual step lengths.
    Min,
    /// Use the maximum of the primal and dual step lengths.
    Max,
    /// Use the arithmetic mean of the primal and dual step lengths.
    Average,
}

impl AlphaForY {
    /// Compute the actual step length for `y_c`, `y_d` given the
    /// already-selected primal and dual step lengths.
    pub fn alpha_y(self, alpha_primal: Number, alpha_dual: Number) -> Number {
        match self {
            AlphaForY::Primal => alpha_primal,
            AlphaForY::BoundMult => alpha_dual,
            AlphaForY::Full => 1.0,
            AlphaForY::Min => alpha_primal.min(alpha_dual),
            AlphaForY::Max => alpha_primal.max(alpha_dual),
            AlphaForY::Average => 0.5 * (alpha_primal + alpha_dual),
        }
    }
}

pub struct BacktrackingLineSearch {
    pub acceptor: Box<dyn BacktrackingLsAcceptor>,
    pub alpha_red_factor: Number,
    /// `alpha_red_factor_min` — the *floor* on one backtracking
    /// reduction, which is what turns the fixed geometric sequence into
    /// a safeguarded interpolation (gh#818). See
    /// [`BacktrackingLineSearch::next_alpha`]. Setting it equal to
    /// `alpha_red_factor` collapses the clamp and restores the plain
    /// `alpha *= alpha_red_factor` sequence.
    ///
    /// The `0.05` here is the *direct-construction* default (tests and
    /// drivers that assemble a line search by hand).
    /// `AlgorithmBuilder::build` overwrites it from
    /// `LineSearchOptions::alpha_red_factor_min`, which resolves to
    /// `0.05` under `limited-memory` and to `alpha_red_factor` — i.e.
    /// off — under an exact Hessian; that field's doc carries the
    /// measurement behind the split.
    pub alpha_red_factor_min: Number,
    pub max_soc: i32,
    /// Threshold for the SOC outer-loop convergence test
    /// `theta_trial <= kappa_soc * theta_soc_old`. Mirrors upstream's
    /// `kappa_soc` (default 0.99).
    pub kappa_soc: Number,
    /// SOC RHS variant. `0` = upstream default ("old"), `1` = scaled
    /// gradient-block variant. Both correspond to upstream's
    /// `soc_method` option.
    pub soc_method: i32,
    /// Number of consecutive shortened iterations before the watchdog
    /// procedure activates. Disabled when `<= 0`. Mirrors upstream's
    /// `watchdog_shortened_iter_trigger` (default 10).
    pub watchdog_shortened_iter_trigger: i32,
    /// Maximum number of outer iterations the watchdog will accept
    /// non-decreasing trial points before reverting to the snapshot.
    /// Mirrors upstream's `watchdog_trial_iter_max` (default 3).
    pub watchdog_trial_iter_max: i32,
    /// Lower bound on α; below this we declare a tiny step (mirrors
    /// `alpha_min_frac` flow, `IpBacktrackingLineSearch.cpp:CalculateAlphaMin`).
    pub alpha_min: Number,
    /// `filter_theta_roundoff_retry` (gh#945) — whether a line search that
    /// has run out of `alpha` at an iterate that is *already feasible to
    /// round-off* gets one more pass with the filter's `theta` axis measured
    /// against `theta`'s own evaluation noise, before the driver hands off to
    /// a restoration phase that has nothing to minimize. `true` is the
    /// default; `false` restores the plain hand-off.
    pub filter_theta_roundoff_retry: bool,
    /// `objective_noise_wolfe` (gh#1016) — whether [`Self::objective_noise_limited`]
    /// can be entered at all. `true` is the default; `false` restores the
    /// plain Armijo/filter test.
    pub objective_noise_wolfe: bool,
    /// Maximum trial-iteration cap before declaring failure.
    pub max_trials: i32,

    // ---- Watchdog state (port of `IpBacktrackingLineSearch.{hpp,cpp}`'s
    //      `in_watchdog_`, `watchdog_iterate_`, `watchdog_delta_`,
    //      `watchdog_alpha_primal_test_`, `watchdog_trial_iter_`,
    //      `watchdog_shortened_iter_`, `last_mu_`).
    //
    // Watchdog mechanism: after `watchdog_shortened_iter_trigger`
    // consecutive shortened (n_steps > 0) accepts, we snapshot the
    // current iterate `(curr, delta, theta, phi, d_phi)` and enter
    // watchdog mode. While in watchdog: the acceptor's reference
    // values are FROZEN to the snapshot for up to
    // `watchdog_trial_iter_max` outer iterations. Each iteration's
    // alpha-loop runs against the frozen reference; if it accepts,
    // watchdog terminates with success ("W"). If it rejects, we
    // accept the last trial anyway (info char 'w') and let the next
    // outer iteration try again. If `watchdog_trial_iter_max` outer
    // iterations all reject, we revert to the snapshot and re-run
    // the alpha-loop on the saved `delta` with `skip_first=true`.
    /// True iff currently inside a watchdog window.
    in_watchdog: bool,
    /// gh#1016: this solve has accepted a step that did not move the iterate
    /// in floating point — the objective's evaluation noise exceeds any
    /// decrease the step could deliver, so the Armijo comparison on `phi` can
    /// no longer tell a good step from a bad one. Sticky for the rest of the
    /// solve; from then on a trial the Armijo/filter test rejects is also
    /// accepted if the *slope* of `phi` along the step at the trial point
    /// passes [`WOLFE_SIGMA`]/[`WOLFE_DELTA`] (the gradient carries no such
    /// noise floor). Never set on a solve whose line searches behave.
    objective_noise_limited: bool,
    /// Set by the α-loop, instead of accepting, when the step it was about to
    /// accept was a numerical null step on a model with no constraint rows;
    /// read by [`Self::run_alpha_loop`], which then re-runs the loop.
    noise_limit_detected: bool,
    /// Snapshot of the iterate at watchdog activation.
    watchdog_iterate: Option<IteratesVector>,
    /// Snapshot of the search direction at watchdog activation.
    watchdog_delta: Option<IteratesVector>,
    /// Number of outer iterations elapsed since watchdog activation.
    watchdog_trial_iter: i32,
    /// Number of consecutive shortened (n_steps > 0) accepts.
    /// Reset on a full step (n_steps == 0), on mu change, on watchdog
    /// success, and on watchdog stop-with-revert.
    watchdog_shortened_iter: i32,
    /// `mu` at the previous outer iteration. A change clears the
    /// watchdog state (`IpBacktrackingLineSearch.cpp:259-270`).
    last_mu: Number,
    /// Frozen reference theta at watchdog activation.
    watchdog_theta: Number,
    /// Frozen reference phi at watchdog activation.
    watchdog_phi: Number,
    /// Frozen reference d_phi at watchdog activation.
    watchdog_d_phi: Number,

    // ---- Soft restoration phase (port of `IpBacktrackingLineSearch`'s
    //      `in_soft_resto_phase_`, `soft_resto_counter_`).
    //
    // When the regular filter line search fails, before handing off to
    // the full (sub-NLP) restoration phase, the driver tries a single
    // damped primal-dual step along the *same* search direction. The
    // step is damped only by the fraction-to-the-boundary rule and is
    // accepted if it either satisfies the original filter criterion
    // ('S' — leave soft resto) or merely reduces the primal-dual KKT
    // system error by `soft_resto_pderror_reduction_factor` ('s' —
    // stay in soft resto). Subsequent outer iterations keep taking
    // soft-resto steps until the original criterion is met, the step
    // is rejected, or `max_soft_resto_iters` consecutive iterations
    // elapse — any of which drops through to full restoration.
    /// Required relative reduction in the primal-dual system error for
    /// a soft-resto step to be accepted. `0` disables soft restoration.
    /// Mirrors upstream `soft_resto_pderror_reduction_factor`
    /// (default `1 - 1e-4`).
    pub soft_resto_pderror_reduction_factor: Number,
    /// Cap on consecutive soft-resto iterations before full
    /// restoration is forced. Mirrors upstream `max_soft_resto_iters`
    /// (default 10).
    pub max_soft_resto_iters: i32,
    /// True iff the driver is currently inside the soft-resto phase.
    in_soft_resto_phase: bool,
    /// Count of consecutive soft-resto iterations taken so far.
    soft_resto_counter: i32,

    /// `accept_every_trial_step` — when true, the alpha loop and filter
    /// are bypassed: the FTB-truncated `alpha_init`/`alpha_dual` step
    /// is set as the trial and accepted unconditionally. Mirrors
    /// upstream's `IpBacktrackingLineSearch.cpp:accept_every_trial_step_`
    /// short-circuit at the top of `FindAcceptableTrialPoint`.
    pub accept_every_trial_step: bool,
    /// `alpha_for_y` policy applied to the equality multipliers `y_c`,
    /// `y_d` when constructing the trial iterate. See [`AlphaForY`].
    pub alpha_for_y: AlphaForY,
    /// `accept_after_max_steps` — once this many backtracking steps have
    /// been taken in one line search, the trial point is accepted
    /// without consulting the acceptor. `-1` (the default, and
    /// upstream's) disables the escape hatch entirely, so the field is
    /// inert unless a caller sets it.
    ///
    /// Port of `IpBacktrackingLineSearch.cpp:759-770`: upstream
    /// evaluates the trial barrier objective and constraint violation
    /// first (so an evaluation error still backtracks — the finiteness
    /// check in the alpha loop is pounce's equivalent), tags the
    /// iteration `MaxS`, and calls `Reset()` — leaving the soft
    /// restoration phase and resetting the acceptor — before accepting.
    ///
    /// Like `accept_every_trial_step`, this drops the global
    /// convergence guarantee: the accepted point satisfies neither the
    /// filter nor the Armijo condition.
    pub accept_after_max_steps: i32,
}

/// Internal alpha-loop outcome. The watchdog wrapper translates this
/// into the public [`Outcome`] after applying its state machine.
enum AlphaResult {
    /// Trial accepted at `alpha_used` after `n_steps` reductions.
    Accepted { n_steps: i32 },
    /// α dropped below `alpha_min_eff` ⇒ tiny step. `last_alpha` is
    /// the smallest α actually evaluated; `n_steps` is the number of
    /// reductions performed.
    TinyStep { n_steps: i32, last_alpha: Number },
    /// `max_trials` exhausted without acceptance. The last attempted
    /// trial iterate is left in `data.trial` so the watchdog
    /// "accept-anyway" path can promote it.
    ///
    /// `evaluation_error` flags that the last attempted trial produced
    /// a non-finite `theta_trial`/`phi_trial` — mirrors upstream's
    /// `evaluation_error` tracked from `IpoptNLP::Eval_Error`
    /// (`IpBacktrackingLineSearch.cpp:776-784`). The watchdog handler
    /// must treat this as a forced StopWatchDog
    /// (`IpBacktrackingLineSearch.cpp:493`) — accepting a non-finite
    /// iterate via the 'w' branch propagates NaN/Inf into the next
    /// outer iter (observed on PFIT3 iter 53: inf_pr=7.87e305 from a
    /// 'w'-accepted trial; on PFIT4 iter 31: inf_pr=1.01e11).
    Failed {
        n_steps: i32,
        last_alpha: Number,
        evaluation_error: bool,
    },
    /// The shared wall/CPU-time deadline was crossed before a trial was
    /// accepted (pounce#242). Propagated up as [`Outcome::Deadline`].
    Deadline,
}

impl BacktrackingLineSearch {
    pub fn new(acceptor: Box<dyn BacktrackingLsAcceptor>) -> Self {
        Self {
            acceptor,
            alpha_red_factor: 0.5,
            alpha_red_factor_min: 0.05,
            max_soc: 4,
            kappa_soc: 0.99,
            soc_method: 0,
            watchdog_shortened_iter_trigger: 10,
            watchdog_trial_iter_max: 3,
            alpha_min: 1e-12,
            filter_theta_roundoff_retry: true,
            objective_noise_wolfe: true,
            max_trials: 50,
            in_watchdog: false,
            objective_noise_limited: false,
            noise_limit_detected: false,
            watchdog_iterate: None,
            watchdog_delta: None,
            watchdog_trial_iter: 0,
            watchdog_shortened_iter: 0,
            last_mu: -1.0,
            watchdog_theta: 0.0,
            watchdog_phi: 0.0,
            watchdog_d_phi: 0.0,
            soft_resto_pderror_reduction_factor: 1.0 - 1e-4,
            max_soft_resto_iters: 10,
            in_soft_resto_phase: false,
            soft_resto_counter: 0,
            accept_every_trial_step: false,
            alpha_for_y: AlphaForY::Primal,
            accept_after_max_steps: -1,
        }
    }

    /// Whether the line search is inside a watchdog trial sequence.
    ///
    /// While this is `true` the iterate in `data.curr` is **provisional**:
    /// it was promoted through the `accept-anyway` branch of
    /// [`Self::handle_watchdog_failure`] (info char `'w'`) despite the
    /// acceptor *rejecting* it, the filter was deliberately not augmented,
    /// and a snapshot of the pre-watchdog iterate and direction is held in
    /// `watchdog_iterate` / `watchdog_delta`. Within
    /// `watchdog_trial_iter_max` (default 3) further iterations the line
    /// search either finds the gamble paid off or executes `StopWatchDog`
    /// and reverts to that snapshot.
    ///
    /// The outer algorithm reads this so a *terminal* verdict is never
    /// pronounced on a point the line search itself has already rejected
    /// and is holding a replacement for — see the divergence guard in
    /// [`crate::ipopt_alg::IpoptAlgorithm::iterate`].
    pub(crate) fn in_watchdog(&self) -> bool {
        self.in_watchdog
    }

    /// Test-only accessor for the shortened-iter counter.
    #[cfg(test)]
    pub(crate) fn watchdog_shortened_iter(&self) -> i32 {
        self.watchdog_shortened_iter
    }

    pub fn acceptor(&self) -> &dyn BacktrackingLsAcceptor {
        &*self.acceptor
    }

    pub fn acceptor_mut(&mut self) -> &mut dyn BacktrackingLsAcceptor {
        &mut *self.acceptor
    }

    /// Reset the acceptor state at the start of a new outer iteration.
    pub fn reset(&mut self) {
        self.acceptor.reset();
    }

    /// Clear the globalization heuristics' cross-iteration counters
    /// after the full restoration phase has *succeeded* — port of
    /// `IpBacktrackingLineSearch.cpp:624-631`.
    ///
    /// Upstream calls `PerformRestoration()` from inside
    /// `FindAcceptableTrialPoint`, so these four assignments sit
    /// directly after it and the state is in scope. pounce hands the
    /// restoration off to the caller (`IpoptAlgorithm::invoke_restoration`)
    /// and returns `Outcome::Failed`, so the reset has to be driven from
    /// there instead — see the `RestorationOutcome::Recovered` arm.
    ///
    /// Getting this wrong is not cosmetic. `watchdog_shortened_iter`
    /// counts *consecutive* shortened steps, and the watchdog arms at
    /// `watchdog_shortened_iter_trigger` (default 10). A restoration
    /// episode is not a shortened step — it is a different point — so
    /// carrying the count across one lets runs of shortened steps that
    /// are separated by restoration accumulate as if they were
    /// consecutive. On `steenbrf` that is exactly what happened: five
    /// shortened steps before restoration plus five after reached the
    /// trigger, the watchdog armed, spent its three trial iterations
    /// and reverted to the pre-watchdog point, and the line search then
    /// collapsed to alpha ~1e-08 with 20+ backtracks. That cycle
    /// repeated 105 times and the solve hit `max_iter`; with the reset
    /// in place the counter never reaches the trigger (upstream
    /// Ipopt's longest run on this problem is 6) and the same
    /// trajectory converges.
    ///
    /// `count_successive_shortened_steps_` (cpp:624) is not ported —
    /// upstream reads it only under `expect_infeasible_problem_`
    /// (cpp:798-804), which pounce does not implement.
    pub fn reset_after_restoration(&mut self) {
        self.in_soft_resto_phase = false;
        self.soft_resto_counter = 0;
        self.watchdog_shortened_iter = 0;
    }

    /// Public line-search entry point. Wraps the regular filter line
    /// search ([`Self::run_filter_line_search`]) with the soft
    /// restoration phase — port of the `in_soft_resto_phase_` state
    /// machine in `IpBacktrackingLineSearch::FindAcceptableTrialPoint`
    /// (`IpBacktrackingLineSearch.cpp:439-465` for the in-phase
    /// continuation, `:528-556` for entering the phase).
    ///
    /// Outcomes:
    /// - `Accepted`: a trial point is in `data.trial` — either a
    ///   regular filter/watchdog step or a soft-resto step (info char
    ///   's' = stay in soft resto, 'S' = step also satisfies the
    ///   original filter so soft resto is left).
    /// - `TinyStep` / `Failed`: neither the regular line search nor a
    ///   soft-resto step could make progress; the caller hands off to
    ///   the full restoration phase.
    #[allow(clippy::too_many_arguments)]
    pub fn find_acceptable_trial_point(
        &mut self,
        data: &IpoptDataHandle,
        cq: &IpoptCqHandle,
        delta: &IteratesVector,
        alpha_init: Number,
        alpha_dual: Number,
        nlp: Option<&Rc<RefCell<dyn IpoptNlp>>>,
        search_dir: Option<&mut PdSearchDirCalc>,
    ) -> Outcome {
        // ---- `accept_every_trial_step` short-circuit. Mirrors the
        // unglobalized path at the top of
        // `IpBacktrackingLineSearch::FindAcceptableTrialPoint` (when
        // `accept_every_trial_step_` is true): no soft-resto, no
        // watchdog, no alpha loop, no filter update — just take the
        // FTB-truncated step (`alpha_init`, `alpha_dual` already
        // include the fraction-to-the-boundary rule) and accept it
        // unconditionally. Used by the Mehrotra cascade.
        if self.accept_every_trial_step {
            let curr = match data.borrow().curr.clone() {
                Some(c) => c,
                None => return Outcome::Failed,
            };
            let alpha_y = self.alpha_for_y.alpha_y(alpha_init, alpha_dual);
            let trial_iv = scaled_step(&curr, delta, alpha_init, alpha_y, alpha_dual);
            let mut d = data.borrow_mut();
            d.set_trial(trial_iv);
            d.info_alpha_primal = alpha_init;
            d.info_alpha_dual = alpha_dual;
            d.info_alpha_primal_char = ' ';
            d.info_ls_count = 1;
            return Outcome::Accepted;
        }

        // ---- Soft-resto continuation. Already inside the phase: bump
        // the counter, bail to full restoration once it exceeds
        // `max_soft_resto_iters`, otherwise take another damped
        // primal-dual step along the caller's `delta`
        // (`IpBacktrackingLineSearch.cpp:439-465`).
        if self.in_soft_resto_phase {
            self.soft_resto_counter += 1;
            if self.soft_resto_counter > self.max_soft_resto_iters {
                self.in_soft_resto_phase = false;
                self.soft_resto_counter = 0;
                return self.fail_to_restoration(data);
            }
            // Per-outer-iteration acceptor hook (no-op for the filter
            // acceptor; the penalty acceptor caches its reference here).
            self.acceptor.init_this_line_search(data, cq, delta);
            return match self.try_soft_resto_step(data, cq, delta) {
                Some(satisfies_original) => {
                    if satisfies_original {
                        self.in_soft_resto_phase = false;
                        self.soft_resto_counter = 0;
                        data.borrow_mut().info_alpha_primal_char = 'S';
                    } else {
                        data.borrow_mut().info_alpha_primal_char = 's';
                    }
                    Outcome::Accepted
                }
                None => {
                    self.in_soft_resto_phase = false;
                    self.soft_resto_counter = 0;
                    self.fail_to_restoration(data)
                }
            };
        }

        // ---- Regular filter line search (watchdog + alpha loop).
        let outcome =
            self.run_filter_line_search(data, cq, delta, alpha_init, alpha_dual, nlp, search_dir);
        if outcome == Outcome::Accepted {
            return Outcome::Accepted;
        }
        // Time budget crossed (pounce#242): the caller is stopping the
        // solve, so skip the soft-restoration attempt and hand the
        // terminal outcome straight back.
        if outcome == Outcome::Deadline {
            return Outcome::Deadline;
        }

        // ---- Regular line search failed. Before the (expensive) full
        // restoration sub-NLP, try to *enter* the soft restoration
        // phase with one damped primal-dual step
        // (`IpBacktrackingLineSearch.cpp:528-556`). `prepare_resto_phase_start`
        // augments the outer filter with the entry envelope — mirrors
        // upstream's `acceptor_->PrepareRestoPhaseStart()` at line 537.
        let reference_theta = cq.borrow().curr_constraint_violation();
        let reference_barr = cq.borrow().curr_barrier_obj();
        self.acceptor
            .prepare_resto_phase_start(reference_theta, reference_barr);
        match self.try_soft_resto_step(data, cq, delta) {
            Some(satisfies_original) => {
                if satisfies_original {
                    data.borrow_mut().info_alpha_primal_char = 'S';
                } else {
                    self.in_soft_resto_phase = true;
                    self.soft_resto_counter = 0;
                    data.borrow_mut().info_alpha_primal_char = 's';
                }
                Outcome::Accepted
            }
            // Soft resto could not help — fall through to full
            // restoration with the original failure outcome. The
            // caller's `invoke_restoration` re-runs
            // `prepare_resto_phase_start`; the duplicate filter
            // augmentation is idempotent (same envelope).
            None => outcome,
        }
    }

    /// Stamp the info fields for a hand-off to the full restoration
    /// phase and return `Outcome::Failed`. Used when the soft
    /// restoration phase exhausts its iteration budget or its step is
    /// rejected mid-phase.
    fn fail_to_restoration(&self, data: &IpoptDataHandle) -> Outcome {
        let mut d = data.borrow_mut();
        d.trial = None;
        d.info_alpha_primal = 0.0;
        d.info_alpha_dual = 0.0;
        d.info_alpha_primal_char = 'R';
        d.info_ls_count = 0;
        Outcome::Failed
    }

    /// Attempt a single damped primal-dual step for the soft
    /// restoration phase — port of
    /// `BacktrackingLineSearch::TrySoftRestoStep`
    /// (`IpBacktrackingLineSearch.cpp:1112-1217`). The step along
    /// `delta` is damped only by the fraction-to-the-boundary rule,
    /// with an identical step length for primal and dual variables.
    ///
    /// Returns:
    /// - `Some(true)`  — trial accepted *and* it satisfies the
    ///   original filter criterion ⇒ caller leaves soft resto ('S').
    /// - `Some(false)` — trial accepted only on the primal-dual error
    ///   reduction test ⇒ caller stays in soft resto ('s').
    /// - `None`        — trial rejected (or soft resto disabled / a
    ///   non-finite evaluation) ⇒ caller falls through to the full
    ///   restoration phase.
    ///
    /// On a `Some(_)` return the accepted trial is left in `data.trial`
    /// and the numeric `info_*` fields are stamped; the caller stamps
    /// `info_alpha_primal_char`.
    fn try_soft_resto_step(
        &mut self,
        data: &IpoptDataHandle,
        cq: &IpoptCqHandle,
        delta: &IteratesVector,
    ) -> Option<bool> {
        // Soft restoration is disabled when the reduction factor is
        // zero (`IpBacktrackingLineSearch.cpp:1124`).
        if self.soft_resto_pderror_reduction_factor == 0.0 {
            return None;
        }
        let curr = data.borrow().curr.clone()?;
        let tau = data.borrow().curr_tau;

        // Identical step length for primal and dual variables, damped
        // only by the fraction-to-the-boundary rule
        // (`IpBacktrackingLineSearch.cpp:1135-1140`).
        let alpha = {
            let cq_ref = cq.borrow();
            cq_ref
                .aff_step_alpha_primal_max(delta, tau)
                .min(cq_ref.aff_step_alpha_dual_max(delta, tau))
        };

        // Soft-resto uses the same scalar α for primal, equality
        // multipliers, and bound multipliers (per upstream).
        let trial_iv = scaled_step(&curr, delta, alpha, alpha, alpha);
        data.borrow_mut().set_trial(trial_iv);

        let theta_trial = cq.borrow().trial_constraint_violation();
        let phi_trial = cq.borrow().trial_barrier_obj();
        if !theta_trial.is_finite() || !phi_trial.is_finite() {
            // Upstream retries up to three times on `Eval_Error`; the
            // step length is fixed, so a non-finite eval here is
            // deterministic — treat it as a rejection.
            return None;
        }

        let theta = cq.borrow().curr_constraint_violation();
        let phi = cq.borrow().curr_barrier_obj();
        let d_phi = self.compute_d_phi(cq, delta);

        // First test: is the trial acceptable to the *original*
        // backtracking globalization? Upstream
        // `acceptor_->CheckAcceptabilityOfTrialPoint(0.)`.
        if self
            .acceptor
            .check_trial_point(0.0, theta, phi, d_phi, theta_trial, phi_trial)
            == AcceptDecision::Accept
        {
            let mut d = data.borrow_mut();
            d.info_alpha_primal = alpha;
            d.info_alpha_dual = alpha;
            d.info_ls_count = 1;
            return Some(true);
        }

        // Second test: sufficient reduction in the primal-dual KKT
        // system error (`IpBacktrackingLineSearch.cpp:1184-1211`).
        let mu = data.borrow().curr_mu;
        let curr_pderror = cq.borrow().curr_primal_dual_system_error(mu);
        let trial_pderror = cq.borrow().trial_primal_dual_system_error(mu);
        if !trial_pderror.is_finite() {
            return None;
        }
        if trial_pderror <= self.soft_resto_pderror_reduction_factor * curr_pderror {
            let mut d = data.borrow_mut();
            d.info_alpha_primal = alpha;
            d.info_alpha_dual = alpha;
            d.info_ls_count = 1;
            return Some(false);
        }
        None
    }

    /// Drive the watchdog state machine + alpha-reduction loop.
    /// Port of `IpBacktrackingLineSearch::FindAcceptableTrialPoint`
    /// (`IpBacktrackingLineSearch.cpp:252-677`) restricted to the
    /// regular (non-soft-resto) filter-acceptor, exact-Hessian path.
    /// The soft restoration phase is layered on top by
    /// [`Self::find_acceptable_trial_point`].
    ///
    /// Outcomes:
    /// - `Accepted`: a trial point is in `data.trial`, info fields are
    ///   stamped. The watchdog state has been advanced (success → "W",
    ///   `accept-anyway` → 'w').
    /// - `TinyStep`: α dropped below the dynamic alpha-min before any
    ///   trial was accepted. Caller hands off to restoration.
    /// - `Failed`: alpha-loop exhausted AND watchdog could not rescue.
    ///   Caller hands off to restoration.
    #[allow(clippy::too_many_arguments)]
    fn run_filter_line_search(
        &mut self,
        data: &IpoptDataHandle,
        cq: &IpoptCqHandle,
        delta: &IteratesVector,
        alpha_init: Number,
        alpha_dual: Number,
        nlp: Option<&Rc<RefCell<dyn IpoptNlp>>>,
        search_dir: Option<&mut PdSearchDirCalc>,
    ) -> Outcome {
        // ---- Watchdog: detect mu change → reset state.
        // Mirrors `IpBacktrackingLineSearch.cpp:259-270`.
        let curr_mu = data.borrow().curr_mu;
        if self.last_mu < 0.0 || self.last_mu != curr_mu {
            self.in_watchdog = false;
            self.watchdog_iterate = None;
            self.watchdog_delta = None;
            self.watchdog_shortened_iter = 0;
            self.last_mu = curr_mu;
        }

        // ---- Watchdog: maybe wake up.
        // Mirrors `IpBacktrackingLineSearch.cpp:376-380`.
        if !self.in_watchdog
            && self.watchdog_shortened_iter_trigger > 0
            && self.watchdog_shortened_iter >= self.watchdog_shortened_iter_trigger
        {
            self.start_watchdog(data, cq, delta);
        }

        // Tell the acceptor how many constraint rows back `theta`'s
        // 1-norm, so its `theta_max` reference can be floored in
        // per-row rather than absolute units. Guarded inside the
        // acceptor to be a no-op once `theta_max` has locked, so this
        // only ever takes effect on the first line search of a solve.
        self.acceptor
            .set_theta_rows(cq.borrow().constraint_violation_rows() as Number);

        // Per-outer-iteration acceptor hook.
        self.acceptor.init_this_line_search(data, cq, delta);

        // Decide reference (theta, phi, d_phi). Mirrors upstream's
        // `FilterLSAcceptor::InitThisLineSearch(in_watchdog)` choice
        // between `curr_*` and the saved `watchdog_*` snapshot.
        let (theta, phi, d_phi) = if self.in_watchdog {
            (self.watchdog_theta, self.watchdog_phi, self.watchdog_d_phi)
        } else {
            let theta = cq.borrow().curr_constraint_violation();
            let phi = cq.borrow().curr_barrier_obj();
            let d_phi = self.compute_d_phi(cq, delta);
            (theta, phi, d_phi)
        };

        // Run the alpha-loop on the caller's `delta`.
        let mut result = self.run_alpha_loop(
            data, cq, delta, alpha_init, alpha_dual, nlp, search_dir, theta, phi, d_phi,
            /*skip_first*/ false, /*theta_floor*/ 0.0,
        );

        // ---- gh#945: the round-off retry.
        //
        // The branch below this one hands a failed line search to the
        // restoration phase, whose job is to *reduce the constraint
        // violation*. When the iterate the hand-off starts from is already
        // feasible to the round-off of its own constraint evaluation, that
        // phase has nothing to minimize: it converges on its first iterate,
        // returns the point it was given, and the solve dies from the
        // optimum with `Error_In_Step_Computation`.
        //
        // What blocked the line search in that situation is not a real
        // filter entry. `theta` at every entry and at every trial is one or
        // two ulp of the same cancellation, so the `theta` arm of
        // `FilterEntry::Acceptable` turns on which way the last constraint
        // sum rounded. Where it rounds the wrong way the entry falls back on
        // its `phi` arm alone, and a one-dimensional filter on `phi` is a
        // *monotone* test — strictly stronger than the Armijo condition the
        // filter method exists to replace, and able to reject the step the
        // algorithm has to take. Measured on gh#945 at iteration 94, trial
        // 3, `alpha = 0.125`: the trial cut `phi` by 5.1e-9 and passed
        // Armijo, the entry's `phi` was 2.0e-8 below it, and the whole
        // decision turned on `theta` 5.55e-16 against 1.11e-16.
        //
        // So before the hand-off that cannot work, re-run the alpha loop
        // once with the `theta` axis measured against `theta`'s own
        // evaluation noise (`IpoptCq::theta_evaluation_noise_floor`). A
        // point that genuinely worsens feasibility is still dominated; a
        // point that differs from an entry only in how the sum rounded is
        // not. If the retry finds nothing either, the original outcome
        // stands and the hand-off proceeds exactly as before — which is why
        // this can only ever be reached on a trajectory that was otherwise
        // about to enter restoration.
        if self.filter_theta_roundoff_retry
            && !self.in_watchdog
            && matches!(
                result,
                AlphaResult::Failed { .. } | AlphaResult::TinyStep { .. }
            )
        {
            // `theta <= floor` is the gate, and it is the whole reason this
            // is a retry rather than a filter setting. Applying the floor to
            // every filter decision fixes gh#945 too and costs MacMPEC's
            // `qpec_small` its answer at exactly the floor gh#945 needs, with
            // no constant and no scale-aware formula between them. This gate
            // is a question about the *iterate*, not about a pair of entries:
            // `qpec_small`'s one line-search failure under the `prod_eq`
            // lowering sits at `theta = 1.746646e-15` against a floor of
            // `6.664715e-16`, 2.6x above it, so restoration there has real
            // violation to work on, the gate declines, and that fixture is
            // byte-identical. See
            // `pounce-algorithm/tests/issue_884_biactive_dual_divergence.rs`,
            // `the_gh945_retry_gate_is_what_this_fixture_relies_on`.
            let floor = cq.borrow().theta_evaluation_noise_floor();
            // The gate reads the same *capped* allowance the filter does, not
            // the raw bound (gh#946 review). `theta_evaluation_noise_floor` is
            // a product of ∞-norms, so on a badly scaled model it runs orders
            // above the noise the near-zero rows carry, and gating on it opens
            // the retry exactly where the rule above says it must not:
            // `square_flowsheet_resto` reads a floor of 2.0e-7 against a
            // `theta` pinned at 1.04e-9, five orders above round-off and with
            // real violation for restoration to work on. The cap is
            // `compare_le`'s band, evaluated here at the iterate as
            // [`super::filter::entry_accepts`] evaluates it at the entry.
            let gate = floor.min(10.0 * Number::EPSILON * theta.abs().max(1.0));
            if floor > 0.0 && theta <= gate {
                // No SOC on the retry: `search_dir` was consumed by the
                // first pass, and the point of this pass is the filter
                // decision, not a different direction.
                let retry = self.run_alpha_loop(
                    data, cq, delta, alpha_init, alpha_dual, nlp, None, theta, phi, d_phi,
                    /*skip_first*/ false, floor,
                );
                // Take the retry's outcome when it found a point, and when
                // the time budget went during it (pounce#242) — that one is
                // terminal and must not be masked by the first pass's
                // failure. Anything else leaves the original outcome, and
                // with it the info fields the hand-off reports.
                if matches!(retry, AlphaResult::Accepted { .. } | AlphaResult::Deadline) {
                    result = retry;
                }
            }
        }

        match result {
            AlphaResult::Accepted { n_steps } => {
                // Update the shortened-iter counter
                // (`IpBacktrackingLineSearch.cpp:644-655`).
                if n_steps == 0 {
                    self.watchdog_shortened_iter = 0;
                } else {
                    self.watchdog_shortened_iter += 1;
                }
                if self.in_watchdog {
                    // Watchdog success — clear state, info char already
                    // stamped by the alpha loop's
                    // `update_for_next_iteration` call. Upstream also
                    // appends "W" to the info string here; pounce
                    // doesn't track an info string yet.
                    self.in_watchdog = false;
                    self.watchdog_iterate = None;
                    self.watchdog_delta = None;
                    self.watchdog_shortened_iter = 0;
                }
                Outcome::Accepted
            }
            AlphaResult::TinyStep {
                n_steps,
                last_alpha,
            } => {
                let mut d = data.borrow_mut();
                d.trial = None;
                d.info_alpha_primal = last_alpha;
                d.info_alpha_dual = 0.0;
                d.info_alpha_primal_char = 'R';
                d.info_ls_count = n_steps + 1;
                Outcome::TinyStep
            }
            AlphaResult::Failed {
                n_steps,
                last_alpha,
                evaluation_error,
            } => {
                if self.in_watchdog {
                    self.handle_watchdog_failure(
                        data,
                        cq,
                        alpha_dual,
                        nlp,
                        n_steps,
                        last_alpha,
                        evaluation_error,
                    )
                } else {
                    // Genuine failure → restoration.
                    let mut d = data.borrow_mut();
                    d.trial = None;
                    d.info_alpha_primal = last_alpha;
                    d.info_alpha_dual = 0.0;
                    d.info_alpha_primal_char = 'R';
                    d.info_ls_count = n_steps + 1;
                    Outcome::Failed
                }
            }
            // Time budget crossed mid-loop (pounce#242) — terminal, and it
            // pre-empts the watchdog: there is no point reverting to a
            // snapshot when the caller is about to stop the solve.
            AlphaResult::Deadline => Outcome::Deadline,
        }
    }

    /// Snapshot the current `(curr, delta, theta, phi, d_phi)` and
    /// activate the watchdog. Mirrors upstream
    /// `IpBacktrackingLineSearch::StartWatchDog`
    /// (`IpBacktrackingLineSearch.cpp:855-869`) plus
    /// `IpFilterLSAcceptor::StartWatchDog`
    /// (`IpFilterLSAcceptor.cpp:506-513`) — pounce stores the
    /// frozen reference values directly on the driver because the
    /// acceptor is stateless w.r.t. reference values (the driver
    /// passes them per call).
    fn start_watchdog(
        &mut self,
        data: &IpoptDataHandle,
        cq: &IpoptCqHandle,
        delta: &IteratesVector,
    ) {
        let curr = data.borrow().curr.clone();
        let Some(curr) = curr else {
            return;
        };
        self.in_watchdog = true;
        self.watchdog_iterate = Some(curr);
        self.watchdog_delta = Some(delta.clone());
        self.watchdog_trial_iter = 0;
        self.watchdog_theta = cq.borrow().curr_constraint_violation();
        self.watchdog_phi = cq.borrow().curr_barrier_obj();
        self.watchdog_d_phi = self.compute_d_phi(cq, delta);
    }

    /// Handle alpha-loop failure while in watchdog mode. Bumps
    /// `watchdog_trial_iter`; if the cap is exceeded, reverts to the
    /// snapshot (StopWatchDog) and re-runs the alpha-loop on the
    /// saved `delta` with `skip_first=true`. Otherwise accepts the
    /// current trial as 'w' and returns. Mirrors
    /// `IpBacktrackingLineSearch.cpp:480-503` together with
    /// `IpBacktrackingLineSearch.cpp:871-908`'s `StopWatchDog`.
    fn handle_watchdog_failure(
        &mut self,
        data: &IpoptDataHandle,
        cq: &IpoptCqHandle,
        alpha_dual: Number,
        nlp: Option<&Rc<RefCell<dyn IpoptNlp>>>,
        n_steps: i32,
        last_alpha: Number,
        evaluation_error: bool,
    ) -> Outcome {
        self.watchdog_trial_iter += 1;
        // Mirror upstream `IpBacktrackingLineSearch.cpp:493`:
        // `if (evaluation_error || watchdog_trial_iter > max)` →
        // StopWatchDog. A non-finite trial must NOT be promoted via
        // the 'w' accept-anyway path; doing so propagates NaN/Inf
        // into the next outer iter and the iterate is unrecoverable
        // (observed on PFIT3, PFIT4).
        if evaluation_error || self.watchdog_trial_iter > self.watchdog_trial_iter_max {
            // StopWatchDog: revert curr to the snapshot, re-run on
            // saved delta with `skip_first=true` (alpha starts at
            // `alpha_init * alpha_red_factor`).
            let snapshot_iter = self.watchdog_iterate.take();
            let snapshot_delta = self.watchdog_delta.take();
            self.in_watchdog = false;
            self.watchdog_shortened_iter = 0;
            let (Some(snap), Some(snap_delta)) = (snapshot_iter, snapshot_delta) else {
                // Defensive — this should not happen if start_watchdog
                // ran successfully. Fall through to genuine failure.
                let mut d = data.borrow_mut();
                d.trial = None;
                d.info_alpha_primal = last_alpha;
                d.info_alpha_dual = 0.0;
                d.info_alpha_primal_char = 'R';
                d.info_ls_count = n_steps + 1;
                return Outcome::Failed;
            };
            {
                let mut d = data.borrow_mut();
                d.set_curr(snap);
            }
            let theta = cq.borrow().curr_constraint_violation();
            let phi = cq.borrow().curr_barrier_obj();
            let d_phi = self.compute_d_phi(cq, &snap_delta);
            // Recompute the fraction-to-the-boundary caps from the
            // *reverted* snapshot direction at the *reverted* iterate
            // (`curr` was just set to `snap`). This mirrors upstream
            // `IpBacktrackingLineSearch::FindAcceptableTrialPoint`, which
            // recomputes `alpha_primal_max` / `alpha_dual_max` from
            // `actual_delta_` after `StopWatchDog` has reverted it to the
            // snapshot — the whole FindAcceptableTrialPoint body re-runs
            // on the recovered direction, caps included.
            //
            // The failed direction's caps (the `alpha_init` / `alpha_dual`
            // this method was handed, sized for the pre-revert iterate and
            // the now-abandoned search direction) are NOT reused: applying
            // them to `snap_delta` is wrong in both directions. If the
            // failed cap is looser than the snapshot's FTB limit, the first
            // retry trial overshoots the boundary — a negative slack /
            // bound-multiplier, i.e. a non-finite barrier objective — and
            // the loop wastes trials backtracking out of infeasibility; if
            // tighter, it needlessly shortens a feasible step. Clamp by the
            // full step `1.0` (the default `alpha_max`), matching the main
            // path's `alpha_init.min(alpha_primal_max)` at
            // `ipopt_alg.rs:1045`.
            let tau = data.borrow().curr_tau;
            let (alpha_primal_retry, alpha_dual_retry) = {
                let cq_ref = cq.borrow();
                (
                    1.0_f64.min(cq_ref.aff_step_alpha_primal_max(&snap_delta, tau)),
                    1.0_f64.min(cq_ref.aff_step_alpha_dual_max(&snap_delta, tau)),
                )
            };
            // SOC is disabled on the StopWatchDog retry. The original
            // `search_dir` was consumed by the first alpha-loop call
            // and we want a plain backtracking pass over the saved
            // delta; mirrors upstream's behavior of not running the
            // soc_method on the recovered search (hence `search_dir =
            // None` and `skip_first = true`, which starts the retry from
            // `alpha_*_retry * alpha_red_factor`).
            let result2 = self.run_alpha_loop(
                data,
                cq,
                &snap_delta,
                alpha_primal_retry,
                alpha_dual_retry,
                nlp,
                None,
                theta,
                phi,
                d_phi,
                /*skip_first*/ true,
                /*theta_floor*/ 0.0,
            );
            match result2 {
                AlphaResult::Accepted { n_steps: ns2 } => {
                    if ns2 == 0 {
                        self.watchdog_shortened_iter = 0;
                    } else {
                        self.watchdog_shortened_iter += 1;
                    }
                    Outcome::Accepted
                }
                AlphaResult::TinyStep {
                    n_steps: ns2,
                    last_alpha: la2,
                } => {
                    let mut d = data.borrow_mut();
                    d.trial = None;
                    d.info_alpha_primal = la2;
                    d.info_alpha_dual = 0.0;
                    d.info_alpha_primal_char = 'R';
                    d.info_ls_count = ns2 + 1;
                    Outcome::TinyStep
                }
                AlphaResult::Failed {
                    n_steps: ns2,
                    last_alpha: la2,
                    evaluation_error: _,
                } => {
                    let mut d = data.borrow_mut();
                    d.trial = None;
                    d.info_alpha_primal = la2;
                    d.info_alpha_dual = 0.0;
                    d.info_alpha_primal_char = 'R';
                    d.info_ls_count = ns2 + 1;
                    Outcome::Failed
                }
                // Deadline crossed during the StopWatchDog retry sweep
                // (pounce#242) — propagate the terminal outcome.
                AlphaResult::Deadline => Outcome::Deadline,
            }
        } else {
            // Accept the last attempted trial despite filter rejection
            // — `accept-anyway` watchdog branch
            // (`IpBacktrackingLineSearch.cpp:498-503`). The trial
            // iterate from the final α attempt is already in
            // `data.trial`. Crucially, we do NOT call
            // `update_for_next_iteration`, so the filter is NOT
            // augmented (matching upstream's char='w' branch at
            // line 833-836 which skips `UpdateForNextIteration`).
            let mut d = data.borrow_mut();
            d.info_alpha_primal = last_alpha;
            d.info_alpha_dual = alpha_dual;
            d.info_alpha_primal_char = 'w';
            d.info_ls_count = n_steps + 1;
            Outcome::Accepted
        }
    }

    /// Inner alpha-reduction loop. Tries
    /// `alpha = alpha_init * alpha_red_factor^k` (or
    /// `alpha_red_factor^(k+1)` when `skip_first=true`) and consults
    /// the acceptor against the supplied reference `(theta, phi, d_phi)`.
    /// On accept stamps the info fields and calls
    /// `update_for_next_iteration`. On reject leaves the LAST trial in
    /// `data.trial` so the watchdog `accept-anyway` path can promote
    /// it.
    #[allow(clippy::too_many_arguments)]
    fn run_alpha_loop(
        &mut self,
        data: &IpoptDataHandle,
        cq: &IpoptCqHandle,
        delta: &IteratesVector,
        alpha_init: Number,
        alpha_dual: Number,
        nlp: Option<&Rc<RefCell<dyn IpoptNlp>>>,
        search_dir: Option<&mut PdSearchDirCalc>,
        theta: Number,
        phi: Number,
        d_phi: Number,
        skip_first: bool,
        theta_floor: Number,
    ) -> AlphaResult {
        // Every α-loop states its `theta` floor, and every α-loop leaves it at
        // zero. Only the gh#945 retry passes a nonzero one, and having the set
        // and the clear be this function's own entry and exit is what makes
        // "left switched on" unreachable rather than merely tested for
        // (gh#946 review): the floor cannot outlive the pass that asked for
        // it, and the paths that read the filter outside an α-loop — the
        // acceptor's `make_orig_progress_check` snapshot among them — always
        // see zero.
        self.acceptor.set_theta_roundoff_floor(theta_floor);
        self.noise_limit_detected = false;
        let mut search_dir = search_dir;
        let mut result = self.run_alpha_loop_inner(
            data,
            cq,
            delta,
            alpha_init,
            alpha_dual,
            nlp,
            search_dir.as_deref_mut(),
            theta,
            phi,
            d_phi,
            skip_first,
        );
        // gh#1016: the loop was about to accept a step too small to move the
        // iterate (and declined to, before touching the acceptor). That is
        // not progress, it is the objective's noise floor swallowing the
        // whole backtracking sweep (each halving changes `phi` by round-off,
        // none of it a decrease). Say so and run the loop once more, now
        // allowed to judge a trial by the slope of `phi` instead of its
        // value. Once per solve: after the flag is set a null step is just a
        // step.
        if self.noise_limit_detected {
            self.objective_noise_limited = true;
            tracing::info!(
                phi,
                d_phi,
                "line search was about to accept a step too small to move the iterate: the \
                 objective's evaluation noise exceeds the decrease it can resolve; judging further trials \
                 by the slope of the barrier objective as well (gh#1016)"
            );
            self.noise_limit_detected = false;
            result = self.run_alpha_loop_inner(
                data,
                cq,
                delta,
                alpha_init,
                alpha_dual,
                nlp,
                search_dir.as_deref_mut(),
                theta,
                phi,
                d_phi,
                skip_first,
            );
        }
        self.acceptor.set_theta_roundoff_floor(0.0);
        result
    }

    /// True iff a step of length `alpha` along `delta` is smaller than the
    /// resolution of the iterate it is added to — `alpha * ||dx||_inf` at or
    /// below a few ulps of `||x||_inf`, so `x + alpha*dx` rounds back to `x`
    /// in the coordinate that moves most.
    fn is_null_step(curr: &IteratesVector, delta: &IteratesVector, alpha: Number) -> bool {
        let moved = alpha * delta.x.amax().max(delta.s.amax());
        let scale = curr.x.amax().max(curr.s.amax());
        scale > 0.0 && moved <= 4.0 * Number::EPSILON * scale
    }

    /// gh#1016: the approximate-Wolfe override. See
    /// [`Self::objective_noise_limited`]. The trial has been staged in
    /// `data.trial`; costs one objective-gradient evaluation.
    fn passes_noise_limited_wolfe(
        &self,
        cq: &IpoptCqHandle,
        delta: &IteratesVector,
        theta: Number,
        theta_trial: Number,
        phi: Number,
        d_phi: Number,
        phi_trial: Number,
    ) -> bool {
        let band = 10.0 * Number::EPSILON;
        if !self.objective_noise_limited
            || self.in_watchdog
            || !(d_phi < 0.0)
            || theta.max(theta_trial) > band
            || (phi_trial - phi).abs() > WOLFE_EPS * phi.abs().max(1.0)
        {
            return false;
        }
        let cq_ref = cq.borrow();
        let d_phi_trial = cq_ref.trial_grad_barrier_obj_x().dot(&*delta.x)
            + cq_ref.trial_grad_barrier_obj_s().dot(&*delta.s);
        d_phi_trial.is_finite()
            && d_phi_trial >= WOLFE_SIGMA * d_phi
            && d_phi_trial <= (2.0 * WOLFE_DELTA - 1.0) * d_phi
    }

    #[allow(clippy::too_many_arguments)]
    fn run_alpha_loop_inner(
        &mut self,
        data: &IpoptDataHandle,
        cq: &IpoptCqHandle,
        delta: &IteratesVector,
        alpha_init: Number,
        alpha_dual: Number,
        nlp: Option<&Rc<RefCell<dyn IpoptNlp>>>,
        search_dir: Option<&mut PdSearchDirCalc>,
        theta: Number,
        phi: Number,
        d_phi: Number,
        skip_first: bool,
    ) -> AlphaResult {
        let curr = match data.borrow().curr.clone() {
            Some(c) => c,
            None => {
                return AlphaResult::Failed {
                    n_steps: 0,
                    last_alpha: 0.0,
                    evaluation_error: false,
                };
            }
        };

        let mut evaluation_error = false;

        let mut soc_search_dir = search_dir;
        let (mut c_soc_buf, mut dms_soc_buf) =
            if soc_search_dir.is_some() && nlp.is_some() && self.max_soc > 0 && !skip_first {
                let cq_ref = cq.borrow();
                let curr_c = cq_ref.curr_c();
                let curr_dms = cq_ref.curr_d_minus_s();
                let mut c_soc = curr_c.make_new();
                c_soc.copy(&*curr_c);
                let mut dms_soc = curr_dms.make_new();
                dms_soc.copy(&*curr_dms);
                (Some(c_soc), Some(dms_soc))
            } else {
                (None, None)
            };

        let mut alpha = if skip_first {
            alpha_init * self.alpha_red_factor
        } else {
            alpha_init
        };
        let mut last_alpha = alpha;
        let mut n_steps: i32 = 0;
        // Smallest step allowed before the loop bails. Upstream
        // `DoBacktrackingLineSearch` sets `alpha_min = alpha_primal_max`
        // (the FTB max step) while in the watchdog window, *bypassing*
        // the acceptor's `CalculateAlphaMin`
        // (`IpBacktrackingLineSearch.cpp:700-704`). Together with the
        // `|| n_steps == 0` loop guard (cpp:740) this guarantees the
        // single full-step watchdog trial always runs, is rejected, and
        // is then routed through the watchdog handler (accept-anyway 'w'
        // or `StopWatchDog` revert). If pounce instead applied the
        // acceptor floor here, a tiny FTB step under watchdog (e.g.
        // scon1dls iter 50, alpha ~6e-13 << acceptor min) would trip the
        // `alpha < alpha_min_eff` early-out below with zero trials and
        // return `TinyStep`, which `run_filter_line_search` hands back
        // directly — bypassing `handle_watchdog_failure`. The watchdog
        // would never revert, `curr` would stay at the diverged iterate,
        // and the solve would die with `ErrorInStepComputation` while
        // upstream IPOPT converges.
        let alpha_min_eff = if self.in_watchdog {
            alpha_init
        } else {
            let acceptor_alpha_min = self.acceptor.calc_alpha_min(d_phi, theta);
            self.alpha_min.max(acceptor_alpha_min)
        };

        for trial in 0..self.max_trials {
            // Fine-grained time-budget gate (pounce#242): each trial
            // evaluates the constraints / barrier objective, which on a
            // large problem is not cheap, so honor the deadline at
            // per-trial granularity rather than letting a full backtracking
            // sweep run past it. Bail before staging another trial; no
            // trial is promoted, so `data.curr` stays the best iterate.
            if data
                .borrow()
                .deadline
                .as_ref()
                .is_some_and(|dl| dl.exceeded().is_some())
            {
                return AlphaResult::Deadline;
            }
            if alpha < alpha_min_eff {
                return AlphaResult::TinyStep {
                    n_steps,
                    last_alpha,
                };
            }
            last_alpha = alpha;
            n_steps = trial;

            let alpha_y = self.alpha_for_y.alpha_y(alpha, alpha_dual);
            let trial_iv = scaled_step(&curr, delta, alpha, alpha_y, alpha_dual);
            data.borrow_mut().set_trial(trial_iv);

            let theta_trial = cq.borrow().trial_constraint_violation();
            let phi_trial = cq.borrow().trial_barrier_obj();
            if !theta_trial.is_finite() || !phi_trial.is_finite() {
                // Mirror upstream `IpBacktrackingLineSearch.cpp:776-784`:
                // a non-finite eval is treated as `Eval_Error`, sets the
                // `evaluation_error` flag, and the alpha-loop continues
                // to backtrack. Under watchdog, upstream breaks out
                // immediately (line 791-794) so the watchdog handler
                // can force StopWatchDog via line 493.
                evaluation_error = true;
                if self.in_watchdog {
                    return AlphaResult::Failed {
                        n_steps: trial,
                        last_alpha: alpha,
                        evaluation_error: true,
                    };
                }
                alpha *= self.alpha_red_factor;
                continue;
            }

            // `accept_after_max_steps` (upstream
            // `IpBacktrackingLineSearch.cpp:759-770`): once this many
            // backtracking steps have been taken, take the point
            // whatever the acceptor thinks of it. Upstream evaluates
            // the trial objective/violation first so an evaluation
            // error still backtracks — that is the finiteness check
            // just above — then calls `Reset()` (leave soft resto,
            // reset the acceptor) and accepts. `-1` disables it, so a
            // solve that does not set the option never takes this
            // branch and the acceptor decides as before.
            let force_accept =
                self.accept_after_max_steps >= 0 && trial >= self.accept_after_max_steps;
            let mut decision = if force_accept {
                self.in_soft_resto_phase = false;
                self.soft_resto_counter = 0;
                self.acceptor.reset();
                AcceptDecision::Accept
            } else {
                self.acceptor
                    .check_trial_point(alpha, theta, phi, d_phi, theta_trial, phi_trial)
            };
            if decision == AcceptDecision::Reject
                && self.passes_noise_limited_wolfe(
                    cq,
                    delta,
                    theta,
                    theta_trial,
                    phi,
                    d_phi,
                    phi_trial,
                )
            {
                decision = AcceptDecision::Accept;
            }
            // gh#1016: a null step is not an acceptance. Checked before the
            // acceptor hears about the step, so the re-run in
            // `run_alpha_loop` starts from an untouched filter. Models with
            // constraint rows are excluded: the slope test reads `phi` alone,
            // and with constraints the multiplier estimate the step leaves
            // behind is not what the slope vouches for (the L-BFGS leg of
            // `eigena2` ends `Solved_To_Acceptable_Level` with an unscaled
            // dual residual of 25 against `dual_inf_tol = 1`).
            if decision == AcceptDecision::Accept
                && self.objective_noise_wolfe
                && !self.objective_noise_limited
                && !self.in_watchdog
                && d_phi < 0.0
                && Self::is_null_step(&curr, delta, alpha)
                && cq.borrow().constraint_violation_rows() == 0
            {
                self.noise_limit_detected = true;
                return AlphaResult::Failed {
                    n_steps: trial,
                    last_alpha: alpha,
                    evaluation_error,
                };
            }
            if decision == AcceptDecision::Accept {
                let mode = self
                    .acceptor
                    .update_for_next_iteration(alpha, theta, phi, d_phi, phi_trial);
                if std::env::var_os("POUNCE_DBG_LS").is_some() {
                    let d = data.borrow();
                    tracing::debug!(target: "pounce::linesearch",
                        "[PN_LS] iter={} mu={:.3e} alpha={:.3e} alpha_d={:.3e} mode={} theta={:.6e} theta_trial={:.6e} phi={:.6e} phi_trial={:.6e} n_steps={}",
                        d.iter_count, d.curr_mu, alpha, alpha_dual, mode, theta, theta_trial, phi, phi_trial, trial
                    );
                }
                let mut d = data.borrow_mut();
                d.info_alpha_primal = alpha;
                d.info_alpha_dual = alpha_dual;
                d.info_ls_count = trial + 1;
                d.info_alpha_primal_char = mode;
                return AlphaResult::Accepted { n_steps: trial };
            }

            // Watchdog: under upstream `IpBacktrackingLineSearch.cpp:791-794`,
            // a failed trial inside the watchdog window breaks out of the
            // alpha-loop immediately — alpha is NOT reduced. The trial just
            // attempted (at the full `alpha_init`) is left in `data.trial`
            // so `handle_watchdog_failure` can promote it via the 'w'
            // accept-anyway branch. Without this break, pounce kept
            // reducing alpha under watchdog and accepted the same tiny
            // step that triggered watchdog activation in the first place,
            // leaving the iterate stalled (observed on HATFLDFLNE: iter 11
            // accepted α=1.22e-4 'h' instead of α=1.00 'w').
            if self.in_watchdog {
                return AlphaResult::Failed {
                    n_steps: trial,
                    last_alpha: alpha,
                    evaluation_error,
                };
            }

            // SOC: only on the first non-skipped trial when constraint
            // violation grew. Disabled when `skip_first=true` (no SOC
            // buffers were allocated). Also disabled under watchdog (the
            // `in_watchdog` break above pre-empts SOC, matching upstream
            // which gates SOC after the in_watchdog break).
            if trial == 0
                && !skip_first
                && self.max_soc > 0
                && theta <= theta_trial
                && c_soc_buf.is_some()
                && dms_soc_buf.is_some()
            {
                let alpha_test = alpha;
                let mut count_soc: i32 = 0;
                let mut theta_soc_old: Number = 0.0;
                let mut theta_trial_local = theta_trial;
                let mut alpha_primal_soc = alpha;
                let mut soc_accepted = false;
                while count_soc < self.max_soc
                    && !soc_accepted
                    && (count_soc == 0 || theta_trial_local <= self.kappa_soc * theta_soc_old)
                {
                    theta_soc_old = theta_trial_local;
                    {
                        let cq_ref = cq.borrow();
                        let trial_c = cq_ref.trial_c();
                        let trial_dms = cq_ref.trial_d_minus_s();
                        if let Some(c_soc) = c_soc_buf.as_mut() {
                            c_soc.scal(alpha_primal_soc);
                            c_soc.axpy(1.0, &*trial_c);
                        }
                        if let Some(dms_soc) = dms_soc_buf.as_mut() {
                            dms_soc.scal(alpha_primal_soc);
                            dms_soc.axpy(1.0, &*trial_dms);
                        }
                    }
                    let delta_soc_opt = {
                        let sd = soc_search_dir
                            .as_deref_mut()
                            .expect("SOC: search_dir is gated above");
                        let nlp_ref = nlp.expect("SOC: nlp is gated above");
                        let c_soc = c_soc_buf.as_deref().expect("SOC: c_soc_buf is gated above");
                        let dms_soc = dms_soc_buf
                            .as_deref()
                            .expect("SOC: dms_soc_buf is gated above");
                        sd.compute_soc_step(
                            data,
                            cq,
                            nlp_ref,
                            c_soc,
                            dms_soc,
                            alpha_primal_soc,
                            self.soc_method,
                        )
                    };
                    let Some(delta_soc) = delta_soc_opt else {
                        break;
                    };
                    let tau = data.borrow().curr_tau;
                    alpha_primal_soc = cq.borrow().aff_step_alpha_primal_max(&delta_soc, tau);
                    // Upstream `IpFilterLSAcceptor.cpp` sets `actual_delta =
                    // delta_soc` on an accepted SOC step: the *entire* step,
                    // primal and dual, is replaced. The dual update therefore
                    // uses the SOC step's own multiplier components — not the
                    // original `delta` — and the dual fraction-to-boundary is
                    // recomputed from `delta_soc`
                    // (`IpBacktrackingLineSearch.cpp:639`). Applying `delta`'s
                    // duals here left the accepted iterate with a primal from
                    // `delta_soc` but duals from `delta`, diverging `inf_du`
                    // from Ipopt on any `H`-flagged iteration (e.g. CRESC4).
                    let alpha_dual_soc = cq.borrow().aff_step_alpha_dual_max(&delta_soc, tau);
                    let mut trial_iv = curr.deep_copy();
                    trial_iv.x.axpy(alpha_primal_soc, &*delta_soc.x);
                    trial_iv.s.axpy(alpha_primal_soc, &*delta_soc.s);
                    trial_iv.y_c.axpy(alpha_primal_soc, &*delta_soc.y_c);
                    trial_iv.y_d.axpy(alpha_primal_soc, &*delta_soc.y_d);
                    trial_iv.z_l.axpy(alpha_dual_soc, &*delta_soc.z_l);
                    trial_iv.z_u.axpy(alpha_dual_soc, &*delta_soc.z_u);
                    trial_iv.v_l.axpy(alpha_dual_soc, &*delta_soc.v_l);
                    trial_iv.v_u.axpy(alpha_dual_soc, &*delta_soc.v_u);
                    let trial_iv = trial_iv.freeze();
                    data.borrow_mut().set_trial(trial_iv);
                    let theta_soc = cq.borrow().trial_constraint_violation();
                    let phi_soc = cq.borrow().trial_barrier_obj();
                    if !theta_soc.is_finite() || !phi_soc.is_finite() {
                        break;
                    }
                    let dec = self
                        .acceptor
                        .check_trial_point(alpha_test, theta, phi, d_phi, theta_soc, phi_soc);
                    if dec == AcceptDecision::Accept {
                        let mode = self
                            .acceptor
                            .update_for_next_iteration(alpha_test, theta, phi, d_phi, phi_soc);
                        let mut d = data.borrow_mut();
                        d.info_alpha_primal = alpha_primal_soc;
                        d.info_alpha_dual = alpha_dual_soc;
                        d.info_ls_count = trial + 1;
                        d.info_alpha_primal_char = mode.to_ascii_uppercase();
                        return AlphaResult::Accepted { n_steps: trial };
                    }
                    count_soc += 1;
                    theta_trial_local = theta_soc;
                    soc_accepted = false;
                }
            }

            alpha = if trial < ALPHA_INTERP_MIN_TRIALS {
                // The fixed sequence gets the first `ALPHA_INTERP_MIN_TRIALS`
                // trials to itself; see that constant for why.
                alpha * self.alpha_red_factor
            } else {
                self.next_alpha(alpha, phi, d_phi, phi_trial)
            };
        }

        AlphaResult::Failed {
            n_steps,
            last_alpha,
            evaluation_error,
        }
    }

    /// The next backtracking trial step, given that `alpha` was
    /// rejected and `phi_trial = φ(alpha)` was measured there.
    ///
    /// Upstream reduces by a fixed factor, `alpha *= alpha_red_factor`
    /// (`IpBacktrackingLineSearch.cpp`), which walks down in halves and
    /// so needs `log₂(1/α*)` trial points to reach a step of size `α*`.
    /// That is cheap when the model is roughly right and ruinous when
    /// it is not: on gh#818's unconstrained ill-conditioned quadratic
    /// the limited-memory model understates the curvature along `d` by
    /// six orders of magnitude, the acceptable step is `α ≈ 4e-6`, and
    /// every iteration spends 19–20 trial points — each a full
    /// objective evaluation — walking there. Under `alpha_red_factor
    /// 0.2` (nine trials instead of twenty) the same solve goes from
    /// `Maximum_Iterations_Exceeded` at 2000 iterations to converged at
    /// 1099, which is the measurement that says the trial *sequence*,
    /// not the acceptance test, is what costs.
    ///
    /// So instead of a fixed factor, fit the quadratic through
    /// `(0, φ)`, `(0, φ')` and `(alpha, φ(alpha))` and jump to its
    /// minimizer — the textbook safeguarded backtracking step
    /// (Nocedal & Wright, *Numerical Optimization* §3.5;
    /// Dennis & Schnabel Alg. A6.3.1). This is a heuristic for *which*
    /// `alpha` to try next and nothing more: the acceptor still decides
    /// whether a trial is taken, so no step this method proposes can be
    /// accepted that the fixed-factor sequence would have rejected.
    ///
    /// Two safeguards keep it bounded, and they are what make the
    /// change safe rather than merely faster:
    ///
    /// * **Never slower than upstream.** The result is capped at
    ///   `alpha_red_factor · alpha`, so the trial sequence still
    ///   contracts at least as fast as the plain geometric one and the
    ///   `alpha < alpha_min_eff` bail is still reached in a bounded
    ///   number of trials.
    /// * **Never a collapse.** The result is floored at
    ///   `alpha_red_factor_min · alpha` (default 0.05, i.e. at most a
    ///   20× reduction per trial), so one badly-shaped `φ` cannot drop
    ///   `alpha` to noise in a single step and skip past an acceptable
    ///   interval. The cap wins if a caller inverts the pair, so the
    ///   two options can be set in either order without aborting the
    ///   solve.
    ///
    /// Falls back to the fixed factor whenever the interpolation is not
    /// defined: a non-descent `d_phi` (the quadratic has no positive
    /// minimizer), a non-finite `phi_trial`, or a non-positive
    /// denominator (`φ(alpha)` below the tangent line, i.e. the fit is
    /// concave and its stationary point is a maximum).
    fn next_alpha(&self, alpha: Number, phi: Number, d_phi: Number, phi_trial: Number) -> Number {
        let fixed = alpha * self.alpha_red_factor;
        if !(d_phi < 0.0) || !phi_trial.is_finite() || !phi.is_finite() {
            return fixed;
        }
        // φ(α) ≈ φ + φ'·α + c·α², with c pinned by the measured
        // `phi_trial`; the minimizer is −φ'/(2c).
        let denom = 2.0 * (phi_trial - phi - d_phi * alpha);
        if !(denom > 0.0) {
            return fixed;
        }
        let alpha_q = -d_phi * alpha * alpha / denom;
        if !alpha_q.is_finite() {
            return fixed;
        }
        // `f64::clamp` panics when `min > max`, and nothing constrains
        // the two options against each other: the registry accepts
        // `alpha_red_factor` anywhere in (0, 1) while the limited-memory
        // default installs `alpha_red_factor_min = 0.05` regardless, so
        // a lone `alpha_red_factor 0.01` is enough to invert them. The
        // cap wins the tie, because "never slower than upstream" is the
        // safeguard that bounds the trial count and the floor is only
        // there to stop a collapse.
        let floor = (alpha * self.alpha_red_factor_min).min(fixed);
        alpha_q.clamp(floor, fixed)
    }

    /// Directional derivative of the barrier objective along the step
    /// `delta`: `d_phi = ∇_x φ · dx + ∇_s φ · ds`.
    fn compute_d_phi(&self, cq: &IpoptCqHandle, delta: &IteratesVector) -> Number {
        let cq_ref = cq.borrow();
        let g_x = cq_ref.curr_grad_barrier_obj_x();
        let g_s = cq_ref.curr_grad_barrier_obj_s();
        g_x.dot(&*delta.x) + g_s.dot(&*delta.s)
    }
}

/// `out = curr + alpha * delta` for all eight components, returned as a
/// fresh `IteratesVector` with `Rc<dyn Vector>` slots. Mirrors
/// `IpoptData::SetTrialBoundMultipliersFromStep` + the primal step
/// path in upstream — both share the same scalar α here because
/// fraction-to-the-boundary truncation has already been folded into
/// `alpha_init` upstream.
fn scaled_step(
    curr: &IteratesVector,
    delta: &IteratesVector,
    alpha_primal: Number,
    alpha_y: Number,
    alpha_dual: Number,
) -> IteratesVector {
    let mut out = curr.make_new_zeroed();
    out.add_one_vector(1.0, curr, 0.0); // out = curr
    out.x.axpy(alpha_primal, &*delta.x);
    out.s.axpy(alpha_primal, &*delta.s);
    out.y_c.axpy(alpha_y, &*delta.y_c);
    out.y_d.axpy(alpha_y, &*delta.y_d);
    out.z_l.axpy(alpha_dual, &*delta.z_l);
    out.z_u.axpy(alpha_dual, &*delta.z_u);
    out.v_l.axpy(alpha_dual, &*delta.v_l);
    out.v_u.axpy(alpha_dual, &*delta.v_u);
    out.freeze()
}

#[cfg(test)]
mod tests {
    use super::*;
    use crate::ipopt_cq::IpoptCalculatedQuantities;
    use crate::ipopt_data::IpoptData;
    use crate::ipopt_nlp::Nlp;
    use crate::iterates_vector::IteratesVector;
    use crate::line_search::filter_acceptor::FilterLsAcceptor;
    use pounce_common::types::Index;
    use pounce_linalg::dense_vector::{DenseVector, DenseVectorSpace};
    use pounce_linalg::expansion_matrix::{ExpansionMatrix, ExpansionMatrixSpace};
    use pounce_linalg::{Matrix, SymMatrix, Vector};
    use std::rc::Rc;

    fn dense(n: i32, vals: &[Number]) -> Rc<dyn Vector> {
        let mut v = DenseVectorSpace::new(n).make_new_dense();
        v.set(0.0);
        if !vals.is_empty() {
            v.values_mut().copy_from_slice(vals);
        }
        Rc::new(v)
    }

    fn dvec(vals: &[Number]) -> DenseVector {
        let mut v = DenseVectorSpace::new(vals.len() as Index).make_new_dense();
        v.set(0.0);
        if !vals.is_empty() {
            v.values_mut().copy_from_slice(vals);
        }
        v
    }

    /// Minimal NLP for the F4 watchdog test: one variable `x[0] >= 0`,
    /// no constraints. `f(x) = x[0]^2`. The only finite bound is the
    /// lower bound on `x[0]`, so the primal fraction-to-the-boundary cap
    /// is governed entirely by the `x[0]` slack.
    struct F4MockNlp {
        x_l: DenseVector,
        x_u: DenseVector,
        d_l: DenseVector,
        d_u: DenseVector,
        px_l: Rc<dyn Matrix>,
        px_u: Rc<dyn Matrix>,
        pd_l: Rc<dyn Matrix>,
        pd_u: Rc<dyn Matrix>,
    }

    impl F4MockNlp {
        fn new() -> Self {
            Self {
                x_l: dvec(&[0.0]),
                x_u: dvec(&[]),
                d_l: dvec(&[]),
                d_u: dvec(&[]),
                // P_L lifts the single lower-bounded var (col 0) into x[0].
                px_l: Rc::new(ExpansionMatrix::new(ExpansionMatrixSpace::new(
                    1,
                    1,
                    &[0],
                    0,
                ))),
                px_u: Rc::new(ExpansionMatrix::new(ExpansionMatrixSpace::new(
                    1,
                    0,
                    &[],
                    0,
                ))),
                pd_l: Rc::new(ExpansionMatrix::new(ExpansionMatrixSpace::new(
                    0,
                    0,
                    &[],
                    0,
                ))),
                pd_u: Rc::new(ExpansionMatrix::new(ExpansionMatrixSpace::new(
                    0,
                    0,
                    &[],
                    0,
                ))),
            }
        }
    }

    impl Nlp for F4MockNlp {
        fn n(&self) -> Index {
            1
        }
        fn m_eq(&self) -> Index {
            0
        }
        fn m_ineq(&self) -> Index {
            0
        }
        fn eval_f(&mut self, x: &dyn Vector) -> Number {
            let xx = x.as_any().downcast_ref::<DenseVector>().unwrap();
            xx.values()[0] * xx.values()[0]
        }
        fn eval_grad_f(&mut self, x: &dyn Vector, g: &mut dyn Vector) {
            let xx = x.as_any().downcast_ref::<DenseVector>().unwrap();
            let gg = g.as_any_mut().downcast_mut::<DenseVector>().unwrap();
            gg.values_mut()[0] = 2.0 * xx.values()[0];
        }
        fn eval_c(&mut self, _x: &dyn Vector, _c: &mut dyn Vector) {}
        fn eval_d(&mut self, _x: &dyn Vector, _d: &mut dyn Vector) {}
        fn eval_jac_c(&mut self, _x: &dyn Vector) -> Rc<dyn Matrix> {
            unimplemented!("no equality constraints in the F4 watchdog fixture")
        }
        fn eval_jac_d(&mut self, _x: &dyn Vector) -> Rc<dyn Matrix> {
            unimplemented!("no inequality constraints in the F4 watchdog fixture")
        }
        fn eval_h(
            &mut self,
            _x: &dyn Vector,
            _obj_factor: Number,
            _y_c: &dyn Vector,
            _y_d: &dyn Vector,
        ) -> Rc<dyn SymMatrix> {
            unimplemented!("Hessian not exercised by the line search")
        }
    }

    impl IpoptNlp for F4MockNlp {
        fn x_l(&self) -> &dyn Vector {
            &self.x_l
        }
        fn x_u(&self) -> &dyn Vector {
            &self.x_u
        }
        fn d_l(&self) -> &dyn Vector {
            &self.d_l
        }
        fn d_u(&self) -> &dyn Vector {
            &self.d_u
        }
        fn px_l(&self) -> Rc<dyn Matrix> {
            self.px_l.clone()
        }
        fn px_u(&self) -> Rc<dyn Matrix> {
            self.px_u.clone()
        }
        fn pd_l(&self) -> Rc<dyn Matrix> {
            self.pd_l.clone()
        }
        fn pd_u(&self) -> Rc<dyn Matrix> {
            self.pd_u.clone()
        }
    }

    /// Acceptor that accepts the first trial unconditionally and records
    /// the primal step it was offered — lets the test read back the
    /// alpha the StopWatchDog retry started from.
    struct RecordingAcceptor {
        first_alpha: Rc<RefCell<Option<Number>>>,
    }

    impl BacktrackingLsAcceptor for RecordingAcceptor {
        fn reset(&mut self) {}
        fn check_trial_point(
            &mut self,
            alpha_primal: Number,
            _theta: Number,
            _phi: Number,
            _d_phi: Number,
            _theta_trial: Number,
            _phi_trial: Number,
        ) -> AcceptDecision {
            let mut slot = self.first_alpha.borrow_mut();
            if slot.is_none() {
                *slot = Some(alpha_primal);
            }
            AcceptDecision::Accept
        }
    }

    fn empty() -> Rc<dyn Vector> {
        dense(0, &[])
    }

    /// F4 (L7 reopen): on the StopWatchDog revert, the alpha-loop retry
    /// must restart from the fraction-to-the-boundary cap of the
    /// *snapshot* direction at the *reverted* iterate — NOT the failed
    /// direction's cap. Pre-fix `handle_watchdog_failure` reused
    /// `alpha_init` (the failed direction's cap); this test pins the
    /// retry's first trial alpha to the recomputed snapshot cap.
    #[test]
    fn stop_watchdog_retry_recomputes_ftb_cap_from_snapshot_direction() {
        let nlp: Rc<RefCell<dyn IpoptNlp>> = Rc::new(RefCell::new(F4MockNlp::new()));
        let data: IpoptDataHandle = Rc::new(RefCell::new(IpoptData::new()));

        // Snapshot iterate: x = 2 (so the x[0] slack is 2), z_L = 0.5.
        let snap = IteratesVector::new(
            dense(1, &[2.0]),
            empty(),
            empty(),
            empty(),
            dense(1, &[0.5]),
            empty(),
            empty(),
            empty(),
        );
        {
            let mut d = data.borrow_mut();
            d.curr_mu = 0.1;
            d.curr_tau = 1.0;
            d.set_curr(snap.clone());
        }
        let cq: IpoptCqHandle = Rc::new(RefCell::new(IpoptCalculatedQuantities::new(
            data.clone(),
            nlp,
        )));

        // Snapshot search direction: Δx = -4. At x = 2 with τ = 1 the
        // fraction-to-the-boundary cap is τ·s/|Δx| = 1·2/4 = 0.5.
        let snap_delta = IteratesVector::new(
            dense(1, &[-4.0]),
            empty(),
            empty(),
            empty(),
            dense(1, &[0.0]),
            empty(),
            empty(),
            empty(),
        );

        let recorded = Rc::new(RefCell::new(None));
        let mut bls = BacktrackingLineSearch::new(Box::new(RecordingAcceptor {
            first_alpha: recorded.clone(),
        }));

        // Arm the watchdog at the snapshot and put it one trial over the
        // cap, so the next failure triggers StopWatchDog (revert + retry).
        bls.in_watchdog = true;
        bls.watchdog_iterate = Some(snap.clone());
        bls.watchdog_delta = Some(snap_delta);
        bls.watchdog_trial_iter = bls.watchdog_trial_iter_max;

        let outcome = bls.handle_watchdog_failure(
            &data, &cq, /*alpha_dual*/ 1.0, None, /*n_steps*/ 0, /*last_alpha*/ 1.0,
            /*evaluation_error*/ false,
        );
        assert_eq!(outcome, Outcome::Accepted);

        // skip_first halves the recomputed cap: 0.5 × alpha_red_factor
        // (0.5) = 0.25. The failed direction's cap would differ.
        let a = recorded
            .borrow()
            .expect("acceptor must have seen at least one trial");
        assert!(
            (a - 0.25).abs() < 1e-12,
            "retry first alpha = {a}, expected 0.25 (snapshot FTB cap 0.5 × red 0.5)"
        );
    }

    /// pounce#242: an already-crossed shared [`Deadline`] on `data` makes
    /// the alpha loop bail on its very first trial with `Outcome::Deadline`
    /// — before staging or evaluating any trial point — so the main loop
    /// can stop the solve at per-trial granularity while `data.curr`
    /// (untouched) remains the best iterate.
    #[test]
    fn deadline_short_circuits_the_alpha_loop() {
        let nlp: Rc<RefCell<dyn IpoptNlp>> = Rc::new(RefCell::new(F4MockNlp::new()));
        let data: IpoptDataHandle = Rc::new(RefCell::new(IpoptData::new()));
        let curr = IteratesVector::new(
            dense(1, &[2.0]),
            empty(),
            empty(),
            empty(),
            dense(1, &[0.5]),
            empty(),
            empty(),
            empty(),
        );
        {
            let mut d = data.borrow_mut();
            d.curr_mu = 0.1;
            d.curr_tau = 1.0;
            d.set_curr(curr.clone());
            // Zero wall budget — already crossed by the time the loop runs.
            d.deadline = Some(pounce_common::timing::Deadline::new(0.0, 1e6));
        }
        let cq: IpoptCqHandle = Rc::new(RefCell::new(IpoptCalculatedQuantities::new(
            data.clone(),
            nlp.clone(),
        )));
        let delta = IteratesVector::new(
            dense(1, &[-1.0]),
            empty(),
            empty(),
            empty(),
            dense(1, &[0.0]),
            empty(),
            empty(),
            empty(),
        );
        let mut bls = BacktrackingLineSearch::new(Box::new(FilterLsAcceptor::new()));
        let outcome = bls.find_acceptable_trial_point(
            &data,
            &cq,
            &delta,
            /*alpha_init*/ 1.0,
            /*alpha_dual*/ 1.0,
            Some(&nlp),
            None,
        );
        assert_eq!(outcome, Outcome::Deadline);
        // No trial was staged/promoted — curr is still the best iterate.
        assert!(data.borrow().trial.is_none());
    }

    fn iv_from(x: &[Number], s: &[Number]) -> IteratesVector {
        IteratesVector::new(
            dense(x.len() as i32, x),
            dense(s.len() as i32, s),
            dense(0, &[]),
            dense(0, &[]),
            dense(0, &[]),
            dense(0, &[]),
            dense(0, &[]),
            dense(0, &[]),
        )
    }

    #[test]
    fn driver_constructs_with_defaults() {
        let bls = BacktrackingLineSearch::new(Box::new(FilterLsAcceptor::new()));
        assert_eq!(bls.alpha_red_factor, 0.5);
        assert_eq!(bls.alpha_red_factor_min, 0.05);
        assert_eq!(bls.max_soc, 4);
    }

    #[test]
    fn scaled_step_writes_curr_plus_alpha_delta() {
        // curr.x = (0,0), delta.x = (1,1) → at alpha=0.5, trial.x = (0.5, 0.5).
        let curr = iv_from(&[0.0, 0.0], &[0.0]);
        let delta = iv_from(&[1.0, 1.0], &[2.0]);
        let trial = scaled_step(&curr, &delta, 0.5, 0.5, 0.5);
        let xv = trial
            .x
            .as_any()
            .downcast_ref::<pounce_linalg::dense_vector::DenseVector>()
            .unwrap()
            .values()
            .to_vec();
        assert_eq!(xv, vec![0.5, 0.5]);
        let sv = trial
            .s
            .as_any()
            .downcast_ref::<pounce_linalg::dense_vector::DenseVector>()
            .unwrap()
            .values()
            .to_vec();
        assert_eq!(sv, vec![1.0]); // 0.0 + 0.5 * 2.0
    }

    #[test]
    fn outcome_variants_are_distinct() {
        assert_ne!(Outcome::Accepted, Outcome::Failed);
        assert_ne!(Outcome::Accepted, Outcome::TinyStep);
        assert_ne!(Outcome::Failed, Outcome::TinyStep);
    }

    #[test]
    fn watchdog_state_starts_inactive() {
        // Mirror upstream `IpBacktrackingLineSearch::InitializeImpl`
        // (`IpBacktrackingLineSearch.cpp:240-249`): the watchdog is
        // inactive at construction and `last_mu_` is initialised to
        // a sentinel `-1` so the first iteration's mu always
        // triggers the reset branch (which is harmless when the
        // watchdog was never armed).
        let bls = BacktrackingLineSearch::new(Box::new(FilterLsAcceptor::new()));
        assert!(!bls.in_watchdog());
        assert_eq!(bls.watchdog_shortened_iter(), 0);
        assert!(bls.last_mu < 0.0);
        assert_eq!(bls.watchdog_shortened_iter_trigger, 10);
        assert_eq!(bls.watchdog_trial_iter_max, 3);
    }

    #[test]
    fn restoration_resets_the_shortened_iter_counter() {
        // Port check for `IpBacktrackingLineSearch.cpp:624-631`. The
        // shortened-iter counter is a *consecutive* count, so a
        // restoration episode has to zero it — otherwise runs of
        // shortened steps on either side of one restoration add up and
        // arm the watchdog where upstream would not.
        //
        // The numbers here are steenbrf's (gh #524): five shortened
        // steps, restoration, five more. Without the reset that is 10 —
        // exactly `watchdog_shortened_iter_trigger` — and the watchdog
        // arms, burns its three trial iterations, reverts, and the line
        // search collapses to alpha ~1e-08. With it the counter tops
        // out at 5 and the solve converges.
        let mut bls = BacktrackingLineSearch::new(Box::new(FilterLsAcceptor::new()));
        bls.watchdog_shortened_iter = 5;
        bls.in_soft_resto_phase = true;
        bls.soft_resto_counter = 4;

        bls.reset_after_restoration();

        assert_eq!(bls.watchdog_shortened_iter, 0);
        assert!(!bls.in_soft_resto_phase);
        assert_eq!(bls.soft_resto_counter, 0);

        // Five more shortened steps after the restoration stay clear of
        // the trigger, which is the whole point.
        bls.watchdog_shortened_iter += 5;
        assert!(bls.watchdog_shortened_iter < bls.watchdog_shortened_iter_trigger);
    }

    #[test]
    fn alpha_result_failed_carries_n_steps_and_last_alpha() {
        // Sanity check on the internal AlphaResult enum: the watchdog
        // wrapper relies on `Failed { n_steps, last_alpha }` to stamp
        // the info-* fields when handing off to restoration.
        let r = AlphaResult::Failed {
            n_steps: 7,
            last_alpha: 1e-6,
            evaluation_error: false,
        };
        match r {
            AlphaResult::Failed {
                n_steps,
                last_alpha,
                evaluation_error,
            } => {
                assert_eq!(n_steps, 7);
                assert!((last_alpha - 1e-6).abs() < 1e-20);
                assert!(!evaluation_error);
            }
            _ => unreachable!(),
        }
    }

    // ---------------------------------------------------- gh#818: next_alpha

    fn ls_for_next_alpha(red: Number, red_min: Number) -> BacktrackingLineSearch {
        let mut bls = BacktrackingLineSearch::new(Box::new(FilterLsAcceptor::default()));
        bls.alpha_red_factor = red;
        bls.alpha_red_factor_min = red_min;
        bls
    }

    /// The interpolated step is what the model says, when the model is
    /// inside the safeguards. `φ(α) = 1 − α + 50α²` has `φ(0) = 1`,
    /// `φ'(0) = −1` and minimizer `1/100`; at the rejected `α = 1`,
    /// `φ(1) = 50`, so the fit is exact and `next_alpha` must return
    /// `0.01` — a 100× reduction the fixed factor would have needed
    /// seven halvings to reach.
    #[test]
    fn next_alpha_jumps_to_the_interpolated_minimizer() {
        let bls = ls_for_next_alpha(0.5, 1e-4);
        let a = bls.next_alpha(1.0, 1.0, -1.0, 50.0);
        assert!((a - 0.01).abs() < 1e-12, "got {a}");
    }

    /// Never slower than upstream: the result is capped at
    /// `alpha_red_factor · alpha`, so a `φ` whose minimizer sits above
    /// the fixed step still contracts at the fixed rate. Without this
    /// cap the `alpha < alpha_min_eff` bail could be pushed arbitrarily
    /// far out, turning a bounded backtracking sweep into
    /// `max_trials` evaluations.
    #[test]
    fn next_alpha_is_never_slower_than_the_fixed_factor() {
        let bls = ls_for_next_alpha(0.5, 1e-4);
        // φ(1) barely above the tangent line ⇒ interpolated minimizer
        // near 1, far above 0.5.
        let a = bls.next_alpha(1.0, 1.0, -1.0, 0.001);
        assert_eq!(a, 0.5, "must clamp up to alpha_red_factor * alpha");
    }

    /// Never a collapse: floored at `alpha_red_factor_min · alpha`, so
    /// one badly-shaped `φ` cannot drop α to noise in a single trial and
    /// step over an acceptable interval.
    #[test]
    fn next_alpha_is_floored_by_alpha_red_factor_min() {
        let bls = ls_for_next_alpha(0.5, 0.05);
        // Minimizer at 1e-6; the floor holds it at 0.05.
        let a = bls.next_alpha(1.0, 1.0, -1.0, 5e5);
        assert!((a - 0.05).abs() < 1e-15, "got {a}");
    }

    /// `alpha_red_factor_min == alpha_red_factor` collapses the clamp,
    /// which is the documented way to restore upstream's fixed
    /// geometric sequence — and the default the builder installs on the
    /// exact-Hessian path.
    #[test]
    fn next_alpha_degenerates_to_the_fixed_factor_when_the_clamp_is_closed() {
        let bls = ls_for_next_alpha(0.5, 0.5);
        for phi_trial in [0.001, 2.0, 50.0, 5e5] {
            assert_eq!(bls.next_alpha(1.0, 1.0, -1.0, phi_trial), 0.5);
        }
    }

    /// The two safeguards can be set in either order, and an inverted
    /// pair must not take the process down. `f64::clamp` panics on
    /// `min > max`, and the pair inverts on one legal option: the
    /// limited-memory default installs `alpha_red_factor_min = 0.05`
    /// while the registry accepts `alpha_red_factor` anywhere in
    /// (0, 1), so `alpha_red_factor 0.01` alone used to abort the solve
    /// mid-iteration (measured on `deb7` at iteration 16). The cap wins
    /// the tie, which also collapses the clamp — an inverted pair gets
    /// upstream's fixed sequence, the same thing a closed clamp gives.
    #[test]
    fn next_alpha_survives_an_inverted_safeguard_pair() {
        // Floor above the cap, both legal on their own.
        let bls = ls_for_next_alpha(0.01, 0.05);
        for phi_trial in [0.001, 2.0, 50.0, 5e5] {
            let a = bls.next_alpha(1.0, 1.0, -1.0, phi_trial);
            assert_eq!(a, 0.01, "inverted pair must fall back to the cap");
        }
        // And the same the other way about, where the clamp is ordinary.
        let bls = ls_for_next_alpha(0.5, 0.05);
        assert!((bls.next_alpha(1.0, 1.0, -1.0, 5e5) - 0.05).abs() < 1e-15);
    }

    /// Every case in which the quadratic fit is not defined falls back
    /// to the fixed factor rather than producing a NaN, a negative step,
    /// or an increase. A NaN α here would be silent: the
    /// `alpha < alpha_min_eff` comparison is false for NaN, so the loop
    /// would keep staging trial points at a NaN step until `max_trials`.
    #[test]
    fn next_alpha_falls_back_on_every_undefined_fit() {
        let bls = ls_for_next_alpha(0.5, 0.05);
        // Non-descent direction: no positive minimizer.
        assert_eq!(bls.next_alpha(1.0, 1.0, 0.0, 2.0), 0.5);
        assert_eq!(bls.next_alpha(1.0, 1.0, 1.0, 2.0), 0.5);
        assert_eq!(bls.next_alpha(1.0, 1.0, Number::NAN, 2.0), 0.5);
        // Non-finite / non-finite-from φ.
        assert_eq!(bls.next_alpha(1.0, 1.0, -1.0, Number::INFINITY), 0.5);
        assert_eq!(bls.next_alpha(1.0, 1.0, -1.0, Number::NAN), 0.5);
        assert_eq!(bls.next_alpha(1.0, Number::NAN, -1.0, 2.0), 0.5);
        // φ(α) at or below the tangent line ⇒ denominator ≤ 0, the fit
        // is concave and its stationary point is a maximum. (The
        // acceptor rejected this α for a filter reason, not an Armijo
        // one, so it is reachable.)
        assert_eq!(bls.next_alpha(1.0, 1.0, -1.0, 0.0), 0.5);
        assert_eq!(bls.next_alpha(1.0, 1.0, -1.0, -5.0), 0.5);
    }
}