ÌÒ×ÓÊÓƵAPP

Speaker: , Argonne National Laboratory

 

Title: Towards McCormick Envelopes for Mixed-integer PDE-constrained Optimization

 

Mixed-integer PDE-constrained optimization (MIPDECO) can be used to model topology optimization and other optimal control application that mix discrete decisions (placement of material) and physical laws modeled by partial-differential equations (PDEs). Unfortunately, many real-world applications result in nonconvex formulations that complicate global convergence guarantees. McCormick envelopes are a well-known standard tool for deriving convex relaxations that can, for example, be used in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the McCormick nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. The resulting approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We prove 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.

This talk does not assume any background in PDEs, and introduces the necessary concepts for our derivation.