McCormick envelopes in mixed-integer PDE-constrained optimization

Journal Article · Mon Jan 06 00:00:00 EST 2025 · Mathematical Programming

DOI:https://doi.org/10.1007/s10107-024-02181-1· OSTI ID:2496737

Leyffer, Sven ^[1]; Manns, Paul ^[2]

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1. Argonne National Laboratory (ANL), Argonne, IL (United States)
2. Technische Univ. of Dortmund (Germany)

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 model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the 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. 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. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

• Research Organization: Argonne National Laboratory (ANL), Argonne, IL (United States)
• Sponsoring Organization: Deutsche Forschungsgemeinschaft (DFG); USDOE; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC)
• Grant/Contract Number: AC02-06CH11357
• Other Award/Contract Number: 540198933; 515118017; 271512359
• OSTI ID: 2496737
• Alternate ID(s): OSTI ID: 3019777
• Journal Information: Mathematical Programming, Journal Name: Mathematical Programming; ISSN 1436-4646; ISSN 0025-5610
• Publisher: Springer
• Country of Publication: United States
• Language: English

Related Subjects

Approximations and Expansions
Calculus of Variations and Optimization
Continuous Optimization
Discrete Optimization
Global Optimization of Nonlinear Mixed-Integer Programming
Optimization
Partial Differential Equations