.. index:: single: lbfgs(Problem)
.. _lbfgs/1:

.. rst-class:: right

**object**

``lbfgs(Problem)``
==================

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


L-BFGS (limited-memory Broyden-Fletcher-Goldfarb-Shanno) quasi-Newton local optimizer with backtracking Armijo line search. Requires the problem to define ``gradient/2``. Supports optional box constraints via projection, minimization and maximization.

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

| **Author:** Paulo Moura
| **Version:** 1:0:0
| **Date:** 2026-09-03

| **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:**

   - Update: Instead of maintaining a dense inverse-Hessian approximation like ``bfgs(_)``, only the last ``memory_size(M)`` step/gradient-difference pairs ``(s, y)`` are kept, and the search direction is recovered from them with the standard two-loop recursion (Nocedal and Wright, Algorithm 7.4). Memory and per-iteration cost are ``O(M*n)`` instead of ``bfgs(_)``'s ``O(n^2)``.
   - Internal minimization form: Maximization is handled by internally minimizing the negated objective and gradient, so the two-loop recursion, curvature test, and Armijo condition are always expressed in minimization form, which avoids sign errors in the line search.
   - Curvature safeguard: Whenever the curvature condition ``y . s > 0`` is not comfortably satisfied (possible here since the line search only enforces sufficient decrease, not a Wolfe curvature condition), the pair history is cleared and the next step falls back to steepest descent, rather than keeping a stale history that would otherwise keep producing the same near-zero-progress direction.
   - Restarts: The ``restart(N)`` option (off by default) periodically clears the pair history, exactly as ``bfgs(_)`` resets its inverse-Hessian approximation to the identity.
   - Bounds: When the problem defines ``position_bounds/1``, trial points are projected onto the box after each step. Projection can weaken the quasi-Newton model; a pure bound-constrained formulation (L-BFGS-B style) 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>`