Bounding-Focused Discretization Methods for the Global Optimization of Nonconvex Semi-Infinite Programs

TL;DR

This paper introduces bounding-focused discretization methods for nonconvex semi-infinite programs, improving convergence efficiency through sensitivity analysis and heuristic algorithms, with demonstrated numerical advantages.

Contribution

It develops a novel bounding-focused discretization framework for SIPs, incorporating sensitivity analysis and heuristics to enhance computational efficiency and convergence.

Findings

Significantly reduces iterations for convergence compared to existing methods.

Ensures solutions of discretizations converge to the true optimal solution.

Demonstrates effectiveness on standard nonconvex SIP test instances.

Abstract

We use sensitivity analysis to design bounding-focused discretization (cutting-surface) methods for the global optimization of nonconvex semi-infinite programs (SIPs). We begin by formulating the optimal bounding-focused discretization of SIPs as a max-min problem and propose variants that are more computationally tractable. We then use parametric sensitivity theory to design an effective heuristic approach for solving these max-min problems. We also show how our new iterative discretization methods may be modified to ensure that the solutions of their discretizations converge to an optimal solution of the SIP. We then formulate optimal bounding-focused generalized discretization of SIPs as max-min problems and design heuristic algorithms for their solution. Numerical experiments on standard nonconvex SIP test instances from the literature demonstrate that our new bounding-focused discretization methods can significantly reduce the number of iterations for convergence relative to a state-of-the-art feasibility-focused discretization method.