.. index:: single: trust_region_newton_cg(Problem)
.. _trust_region_newton_cg/1:

.. rst-class:: right

**object**

``trust_region_newton_cg(Problem)``
===================================

* ``Problem`` - Problem object implementing ``local_optimization_problem_protocol`` and defining ``gradient/2`` and ``hessian/2``.


Trust-region Newton-CG local optimizer (Steihaug-CG for the subproblem). Requires the problem to define ``gradient/2`` and ``hessian/2``. Supports optional box constraints via projection, minimization and maximization.

| **Availability:** 
|    ``logtalk_load(local_optimization(loader))``

| **Author:** Paulo Moura
| **Version:** 1:0:0
| **Date:** 2026-08-24

| **Compilation flags:**
|    ``static, context_switching_calls``


| **Imports:**
|    ``public`` :ref:`local_optimization_solver(Problem) <local_optimization_solver/1>`
| **Uses:**
|    :ref:`linear_algebra <linear_algebra/0>`
|    :ref:`list <list/0>`

| **Remarks:**

   - Subproblem: At each outer iteration, the step is obtained by approximately minimizing the local quadratic model within a ball of radius ``trust_radius``, using the Steihaug-CG method (Nocedal and Wright, Algorithm 7.2): plain conjugate gradient on the model, terminated early either by a negative-curvature direction or by reaching the trust-region boundary, in which case the step is extended to the boundary along the current CG direction.
   - No line search: Unlike the other gradient-based solvers in this library, this solver never backtracks a step size; the trust-region radius itself is grown or shrunk each iteration based on how well the quadratic model predicted the actual objective change, and a step is accepted only when that agreement is good enough.
   - Internal minimization form: Maximization is handled by internally minimizing the negated objective, gradient, and Hessian, so the subproblem and acceptance test are always expressed in minimization form, which avoids sign errors.
   - Convergence: Because it uses exact second-order information, this solver typically converges in far fewer iterations than ``gradient_descent(_)``, ``bfgs(_)``, or ``lbfgs(_)`` on well-behaved problems, at the cost of requiring an explicit ``hessian/2``.
   - Bounds: When the problem defines ``position_bounds/1``, trial points are projected onto the box after each step. Projection can weaken the trust-region model agreement (the accepted step may differ from the one the subproblem solved for), which can trigger more radius shrinkage than an unconstrained problem would; a pure bound-constrained formulation is not implemented.

| **Inherited public predicates:**
|     :ref:`options_protocol/0::check_option/1`  :ref:`options_protocol/0::check_options/1`  :ref:`options_protocol/0::default_option/1`  :ref:`options_protocol/0::default_options/1`  :ref:`options_protocol/0::option/2`  :ref:`options_protocol/0::option/3`  :ref:`local_optimization_solver/1::run/2`  :ref:`local_optimization_solver/1::run/3`  :ref:`local_optimization_solver/1::run/4`  :ref:`options_protocol/0::valid_option/1`  :ref:`options_protocol/0::valid_options/1`  

.. contents::
   :local:
   :backlinks: top

Public predicates
-----------------

(no local declarations; see entity ancestors if any)

Protected predicates
--------------------

(no local declarations; see entity ancestors if any)

Private predicates
------------------

(no local declarations; see entity ancestors if any)

Operators
---------

(none)

.. seealso::

   :ref:`local_optimization_problem_protocol <local_optimization_problem_protocol/0>`, :ref:`local_optimization_solver(Problem) <local_optimization_solver/1>`, :ref:`gradient_descent(Problem) <gradient_descent/1>`, :ref:`conjugate_gradient(Problem) <conjugate_gradient/1>`, :ref:`bfgs(Problem) <bfgs/1>`, :ref:`lbfgs(Problem) <lbfgs/1>`