`bfgs(Problem)`

Problem - Problem object implementing local_optimization_problem_protocol and defining gradient/2. BFGS (Broyden-Fletcher-Goldfarb-Shanno) dense 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: Maintains a dense approximation to the inverse Hessian, updated after every accepted step with the standard BFGS rank-two formula. The approximation starts at the identity matrix.
- Internal minimization form: Maximization is handled by internally minimizing the negated objective and gradient; the quasi-Newton direction, curvature test, and Armijo condition are therefore 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 inverse-Hessian approximation is reset to the identity matrix and the next step falls back to steepest descent, rather than keeping a stale approximation that would otherwise keep producing the same near-zero-progress direction.
- Restarts: The restart(N) option (off by default) periodically resets the inverse-Hessian approximation to the identity matrix, exactly as conjugate_gradient(_) resets its search direction.
- 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: | Â :ref:`options_protocol/0::check_option/1` Â :ref:`options_protocol/0::check_options/1` Â :ref:`options_protocol/0::default_option/1` Â :ref:`options_protocol/0::default_options/1` Â :ref:`options_protocol/0::option/2` Â :ref:`options_protocol/0::option/3` Â :ref:`local_optimization_solver/1::run/2` Â :ref:`local_optimization_solver/1::run/3` Â :ref:`local_optimization_solver/1::run/4` Â :ref:`options_protocol/0::valid_option/1` Â :ref:`options_protocol/0::valid_options/1` Â

.. 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>`