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.. default-domain:: cpp

==========
Optimizers
==========

This chapter describes optimizers implemented in |short_name|.

.. _alg_newton_cg:

===================
Newton-CG Optimizer
===================

.. include::  ../../../includes/optimizers/newton-cg-introduction.rst

------------------------
Mathematical Formulation
------------------------

.. _newton_cg_c_math:

Computing
---------

The Newton-CG optimizer, also known as the hessian-free optimizer, minimizes convex functions 
without calculating the Hessian matrix. Instead, it uses a Hessian product matrix operator.
In the Newton method, the descent direction is calculated using the formula :math:`d_k = -H_k^{-1} g_k`,
where :math:`g_k, H_k` are the gradient and hessian matrix of the loss function on the :math:`k`-th iteration. 
The Newton-CG method uses the Conjugate Gradients solver to find the approximate solution to the equation :math:`H_k d_k = -g_k`.
This solver can find solutions to the system of linear equations :math:`Ax = b` taking 
vector :math:`b` and functor :math:`f(p) = Ap` as input.  


.. _newton_cg_c_math_dense:

Computation Method: *dense*
---------------------------
The method defines the Newton-CG optimizer used by other algorithms
for convex optimization. There is no separate computation mode to minimize a
function manually.

---------------------
Programming Interface
---------------------

Refer to :ref:`API Reference: Newton-CG optimizer <api_newton_cg>`.