A Sequential Homotopy Method for Mathematical Programming Problems
Andreas Potschka, Hans Georg Bock
TL;DR
The paper develops a sequential homotopy framework for constrained optimization in infinite-dimensional Hilbert spaces by coupling a primal–dual projected gradient/antigradient flow of the augmented Lagrangian $L^{\rho}$ with projected backward Euler timestepping. Equilibria of the projected flow correspond to critical points of the original problem, and under GCQ, non-minimizing critical points fail to be asymptotically attracting, enabling robust globalization. The method constructs a sequence of proximally regularized subproblems whose solutions are obtained via a semismooth Newton approach, with continuation in the homotopy parameter yielding global convergence for challenging, badly conditioned PDE-constrained problems. Compared to traditional VI/SQP approaches, this sequential homotopy strategy guarantees feasibility of subproblems, enforces a strong constraint qualification, and effectively globalizes locally convergent solvers. Numerical results on nonlinear, badly conditioned elliptic PDE-constrained optimal control problems demonstrate mesh-independent convergence and superior performance of the proposed approach relative to standard nonlinear VI solvers.
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.