The result of reducing a tight active face to a minimal independent set — the
shared output of the ConstraintSet reduced-face op (Dense arm =
dense_reduced_face; KhatriRaoCone / BlockDiagonal arms produce the same
shape). Determinism: representatives are the lowest-flat-index row per
independent direction, ascending, with no float tie-break.
Primal-feasibility tolerance the inequality-constrained active-set Newton
solver guarantees on its returned iterate, measured in the unit-normalized
constraint-row metric (b_i − a_i·β)/‖a_i‖ that
ConstraintSet::max_scaled_violation computes.
The shared tight-face reduction op over the ConstraintSet carrier union.
An extension trait (not an inherent method) so the numeric reduction
stays in gam-solve where the solvers consume it, keeping gam-problem a pure
data crate. All three arms produce the same ReducedFace contract.
Dense arm of the reduced-face op: reduce the tight rows of an explicit
A x ≥ b set at beta to a minimal independent set. Mirrors
khatri_rao_cone_reduced_face exactly — ascending-index greedy MGS,
RANK_ALPHA·ε·max(n_tight,p)·max‖a‖ tolerance, (A)-strict parallel-only
dependence (|cos| ≥ 1−1e-9, coeff = a_depᵀa_rep/‖a_rep‖², row = the
dependent row’s flat id) — so both carriers produce the same ReducedFace
contract. Flat id = the constraint row index. A zero-norm row is vacuous
(never a direction, never a representative).
Reduce the tight active face of a Khatri–Rao monotonicity cone to its minimal
independent set — the KhatriRaoCone arm of the ConstraintSet reduced-face
op (gam#2306; the Dense arm is dense_reduced_face).
Project point to a strictly interior feasible point of the polyhedron
{β : A·β ≥ b}: the solution of min_β ½‖β − point‖² subject to the
margin-shifted system A·β ≥ b + δ·‖a_i‖, with δ = ACTIVE_SET_INTERIOR_SEED_MARGIN.
Strictly-interior projection onto a ConstraintSet: the operator
analogue of project_point_strictly_into_feasible_cone. Dense sets
delegate to the dense projection (including its anti-parallel equality
lift); the factored cone is homogeneous and one-sided, so the projection
is a single identity-Hessian QP against the margin-shifted rows. The QP uses
the same finite dual active-set solver as production metric projection:
coefficient dimension bounds its dense work, while carrier row count enters
only through operator scans.