object

`lbfgs_b(Problem)`

Problem - Problem object implementing local_optimization_problem_protocol with gradient/2 (and position_bounds/1 when box constraints are present).

L-BFGS-B bound-constrained limited-memory quasi-Newton optimizer with a level-B approximate generalized Cauchy point (first-segment quadratic min along the projected gradient path), free-set identification at the Cauchy point, feasible-step limiting, and L-BFGS two-loop recursion. Requires gradient/2.

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 local_optimization_solver(Problem)

Uses:

linear_algebra

list

Remarks:

Algorithm: Each iteration (1) builds an approximate generalized Cauchy point by minimizing a quadratic model of the limited-memory BFGS Hessian along the first segment of the projected steepest-descent path, (2) identifies the free set at that point, (3) computes an L-BFGS direction via the two-loop recursion and zeroes components outside the free set or that would leave the box, (4) uses the largest feasible step as the Armijo upper bound, and (5) updates the pair history only when the curvature condition holds.

Approximate GCP: Level-B approximation: breakpoints along x(t) = P(x - t g), univariate quadratic minimization on [0, t1] (first breakpoint), with curvature d'Bd estimated from the most recent L-BFGS pair (gamma). Full multi-segment BLNZ Cauchy search is not implemented.

Versus lbfgs(_): Plain lbfgs(_) only clamps trial points after an unconstrained step. This solver never proposes an infeasible step, stops on the projected gradient norm, and chooses the free set from an approximate Cauchy point.

Internal minimization form: Maximization is handled by minimizing the negated objective and gradient (phi-space), as in bfgs(_) and lbfgs(_).

Unbounded problems: When the problem does not define position_bounds/1, the solver behaves like unconstrained lbfgs(_) (no GCP / free-set masking). Prefer lbfgs(_) for purely unconstrained work; prefer this solver when box constraints are present.