A note on semi-infinite program bounding methods

TL;DR

This paper examines semi-infinite program bounding methods, providing a counterexample to a previous claim about convergence, and clarifies the conditions needed for approximate solutions to ensure correct bounds.

Contribution

It refutes a prior claim on convergence of a bounding method and clarifies the conditions for approximate solutions to maintain correctness.

Findings

Counterexample disproves the previous convergence claim.

Conditions for approximate solutions to ensure correct bounds are identified.

Discussion on the limitations of existing bounding methods.

Abstract

Semi-infinite programs are a class of mathematical optimization problems with a finite number of decision variables and infinite constraints. As shown by Blankenship and Falk (Blankenship and Falk. "Infinitely constrained optimization problems." Journal of Optimization Theory and Applications 19.2 (1976): 261-281.), a sequence of lower bounds which converges to the optimal objective value may be obtained with specially constructed finite approximations of the constraint set. In (Mitsos. "Global optimization of semi-infinite programs via restriction of the right-hand side." Optimization 60.10-11 (2011): 1291-1308.), it is claimed that a modification of this lower bounding method involving approximate solution of the lower-level program yields convergent lower bounds. We show with a counterexample that this claim is false, and discuss what kind of approximate solution of the lower-level program is sufficient for correct behavior.