Continuation method for PDE-constrained global optimization: Analysis and application to the shallow water equations

TL;DR

This paper introduces a continuation method using nested homotopies to solve non-convex PDE-constrained optimization problems globally, demonstrated on shallow water equations for control of large-scale systems.

Contribution

It develops a novel interior point continuation approach with a nested homotopy framework for global solutions to PDE-constrained non-convex optimization problems.

Findings

Method successfully solves large-scale PDE-constrained problems.

Applicable to real-world control of cyber-physical systems.

Demonstrates effectiveness on shallow water equations.

Abstract

This paper shows how a class of non-convex optimization problems constrained by discretized nonlinear partial differential equations may be solved to global optimality using an interior point continuation method. The solution procedure rests on a nested homotopy. The inner homotopy solves a barrier problem by driving the barrier parameter to zero. The outer homotopy deforms a convex relaxation to the original non-convex problem in a way that stays clear of bifurcations. A requirement for global optimality is that the objective is convex and that the search space remains path-connected. As a case study, a class of real-world optimization problems subject to the shallow water equations is analyzed. A benchmark as well as a practical implementation demonstrate that the approach is suitable for closed-loop non-convex model predictive control of large-scale cyber-physical systems.