A note on generalized semi-infinite program bounding methods

TL;DR

This paper critically examines a proposed global optimization method for generalized semi-infinite programs, revealing a counterexample that challenges the claim of convergence of the lower bounds to the true optimal value.

Contribution

It provides a counterexample demonstrating that the existing lower bounding method does not always converge to the true optimal value in GSIP.

Findings

Counterexample disproves the convergence claim

Existing method may produce non-converging bounds

Highlights need for revised optimization approaches

Abstract

Generalized semi-infinite programs (GSIP) are a class of mathematical optimization problems that generalize semi-infinite programs, which have a finite number of decision variables and infinite constraints. Mitsos et al. (Mitsos and Tsoukalas. "Global optimization of generalized semi-infinite programs via restriction of the right hand side." Journal of Global Optimization 61.1 (2015): 1-17.) present a method for global optimization of GSIP. This method involves a lower bounding method, and they claim that these lower bounds converge to the optimal objective value of the GSIP. A counterexample is presented that shows that this claim is false.