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```
```// Copyright 2018-2020 argmin developers
//
// Licensed under the Apache License, Version 2.0 <LICENSE-APACHE or
// http://opensource.org/licenses/MIT>, at your option. This file may not be
// copied, modified, or distributed except according to those terms.

//! TODO: Stop when search direction is close to 0
//!
//! # References:
//!
//! [0] Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization.
//! Springer. ISBN 0-387-30303-0.

use crate::prelude::*;
use serde::{Deserialize, Serialize};

/// The Newton-CG method (also called truncated Newton method) uses a modified CG to solve the
/// Newton equations approximately. After a search direction is found, a line search is performed.
///
/// [Example](https://github.com/argmin-rs/argmin/blob/master/examples/newton_cg.rs)
///
/// # References:
///
/// [0] Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization.
/// Springer. ISBN 0-387-30303-0.
#[derive(Clone, Serialize, Deserialize)]
pub struct NewtonCG<L, F> {
/// line search
linesearch: L,
/// curvature_threshold
curvature_threshold: F,
/// Tolerance for the stopping criterion based on cost difference
tol: F,
}

impl<L, F: ArgminFloat> NewtonCG<L, F> {
/// Constructor
pub fn new(linesearch: L) -> Self {
NewtonCG {
linesearch,
curvature_threshold: F::from_f64(0.0).unwrap(),
tol: F::epsilon(),
}
}

/// Set curvature threshold
pub fn curvature_threshold(mut self, threshold: F) -> Self {
self.curvature_threshold = threshold;
self
}

/// Set tolerance for the stopping criterion based on cost difference
pub fn with_tol(mut self, tol: F) -> Result<Self, Error> {
if tol <= F::from_f64(0.0).unwrap() {
return Err(ArgminError::InvalidParameter {
text: "Newton-CG: tol must be positive.".to_string(),
}
.into());
}
self.tol = tol;
Ok(self)
}
}

impl<O, L, F> Solver<O> for NewtonCG<L, F>
where
O: ArgminOp<Output = F, Float = F>,
O::Param: Send
+ Sync
+ Clone
+ Serialize
+ Default
+ ArgminSub<O::Param, O::Param>
+ ArgminDot<O::Param, O::Float>
+ ArgminScaledAdd<O::Param, O::Float, O::Param>
+ ArgminMul<F, O::Param>
+ ArgminConj
+ ArgminZeroLike
+ ArgminNorm<O::Float>,
O::Hessian: Send
+ Sync
+ Default
+ Clone
+ Serialize
+ Default
+ ArgminInv<O::Hessian>
+ ArgminDot<O::Param, O::Param>,
L: Clone + ArgminLineSearch<O::Param, O::Float> + Solver<OpWrapper<O>>,
F: ArgminFloat + Default + ArgminDiv<O::Float, O::Float> + ArgminNorm<O::Float> + ArgminConj,
{
const NAME: &'static str = "Newton-CG";

fn next_iter(
&mut self,
op: &mut OpWrapper<O>,
state: &IterState<O>,
) -> Result<ArgminIterData<O>, Error> {
let param = state.get_param();
let hessian = op.hessian(&param)?;

// Solve CG subproblem
let cg_op: CGSubProblem<O::Param, O::Hessian, O::Float> =
CGSubProblem::new(hessian.clone());
let mut cg_op = OpWrapper::new(cg_op);

let mut x_p = param.zero_like();
let mut x: O::Param = param.zero_like();

let mut cg_state = IterState::new(x_p.clone());
cg.init(&mut cg_op, &cg_state)?;
for iter in 0.. {
let data = cg.next_iter(&mut cg_op, &cg_state)?;
x = data.get_param().unwrap();
let p = cg.p_prev();
let curvature = p.dot(&hessian.dot(&p));
if curvature <= self.curvature_threshold {
if iter == 0 {
break;
} else {
x = x_p;
break;
}
}
if data.get_cost().unwrap()
{
break;
}
cg_state.param(x.clone());
cg_state.cost(data.get_cost().unwrap());
x_p = x.clone();
}

// perform line search
self.linesearch.set_search_direction(x);

// Run solver
let ArgminResult {
operator: line_op,
state:
IterState {
param: next_param,
cost: next_cost,
..
},
} = Executor::new(
OpWrapper::new_from_wrapper(op),
self.linesearch.clone(),
param,
)
.cost(state.get_cost())
.ctrlc(false)
.run()?;

op.consume_op(line_op);

Ok(ArgminIterData::new().param(next_param).cost(next_cost))
}

fn terminate(&mut self, state: &IterState<O>) -> TerminationReason {
if (state.get_cost() - state.get_prev_cost()).abs() < self.tol {
TerminationReason::NoChangeInCost
} else {
TerminationReason::NotTerminated
}
}
}

#[derive(Clone, Default, Serialize, Deserialize)]
struct CGSubProblem<T, H, F> {
hessian: H,
phantom: std::marker::PhantomData<T>,
float: std::marker::PhantomData<F>,
}

impl<T, H, F> CGSubProblem<T, H, F>
where
T: Clone + Send + Sync,
H: Clone + Default + ArgminDot<T, T> + Send + Sync,
{
/// constructor
pub fn new(hessian: H) -> Self {
CGSubProblem {
hessian,
phantom: std::marker::PhantomData,
float: std::marker::PhantomData,
}
}
}

impl<T, H, F> ArgminOp for CGSubProblem<T, H, F>
where
T: Clone + Default + Send + Sync + Serialize + serde::de::DeserializeOwned,
H: Clone + Default + ArgminDot<T, T> + Send + Sync + Serialize + serde::de::DeserializeOwned,
F: ArgminFloat,
{
type Param = T;
type Output = T;
type Hessian = ();
type Jacobian = ();
type Float = F;

fn apply(&self, p: &T) -> Result<T, Error> {
Ok(self.hessian.dot(&p))
}
}

#[cfg(test)]
mod tests {
use super::*;
use crate::solver::linesearch::MoreThuenteLineSearch;
use crate::test_trait_impl;

test_trait_impl!(
newton_cg,
NewtonCG<MoreThuenteLineSearch<Vec<f64>, f64>, f64>
);

test_trait_impl!(cg_subproblem, CGSubProblem<Vec<f64>, Vec<Vec<f64>>, f64>);

#[test]
fn test_tolerance() {
let tol1: f64 = 1e-4;

let linesearch: MoreThuenteLineSearch<Vec<f64>, f64> = MoreThuenteLineSearch::new();

let NewtonCG { tol: t, .. }: NewtonCG<MoreThuenteLineSearch<Vec<f64>, f64>, f64> =
NewtonCG::new(linesearch).with_tol(tol1).unwrap();

assert!((t - tol1).abs() < std::f64::EPSILON);
}
}
```