A Sequential Homotopy Method for Mathematical Programming Problems

TL;DR

This paper introduces a sequential homotopy method for solving complex mathematical programming problems in Hilbert spaces, ensuring feasibility, uniqueness, and convergence while effectively avoiding saddle points and maxima.

Contribution

It develops a novel homotopy approach that guarantees feasibility, uniqueness, and convergence for abstract optimization problems, extending the applicability of local methods.

Findings

Method guarantees feasibility and uniqueness of solutions.

Effectively avoids convergence to saddle points and maxima.

Demonstrated efficiency on nonlinear control problems.

Abstract

We propose a sequential homotopy method for the solution of mathematical programming problems formulated in abstract Hilbert spaces under the Guignard constraint qualification. The method is equivalent to performing projected backward Euler timestepping on a projected gradient/antigradient flow of the augmented Lagrangian. The projected backward Euler equations can be interpreted as the necessary optimality conditions of a primal-dual proximal regularization of the original problem. The regularized problems are always feasible, satisfy a strong constraint qualification guaranteeing uniqueness of Lagrange multipliers, yield unique primal solutions provided that the stepsize is sufficiently small, and can be solved by a continuation in the stepsize. We show that equilibria of the projected gradient/antigradient flow and critical points of the optimization problem are identical, provide sufficient conditions for the existence of global flow solutions, and show that critical points with emanating descent curves cannot be asymptotically stable equilibria of the projected gradient/antigradient flow, practically eradicating convergence to saddle points and maxima. The sequential homotopy method can be used to globalize any locally convergent optimization method that can be used in a homotopy framework. We demonstrate its efficiency for a class of highly nonlinear and badly conditioned control constrained elliptic optimal control problems with a semismooth Newton approach for the regularized subproblems. In contrast to the published article, this version contains a correction that the associate editor considers as too insignificant to justify publication in the journal.