McCormick envelopes in mixed-integer PDE-constrained optimization

TL;DR

This paper explores the application of McCormick envelopes to PDE-constrained optimization problems, proposing a discretized approximation approach that provides convex relaxations with quantifiable error bounds and convergence properties.

Contribution

It introduces a novel discretization method for McCormick relaxations in PDE problems, analyzes their convergence, and demonstrates potential through computational examples.

Findings

Convex relaxations underestimate the original problem with mesh-dependent error bounds.

Approximate relaxations can be improved via bound-tightening procedures.

Minimizers of relaxations converge to those of a limit problem as mesh size approaches zero.

Abstract

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed.