Semi-infinite Programming

Abstract

Semi-infinite programs are constrained optimization problems in which the number of decision variables is finite, but the number of constraints is infinite. In this chapter, we treat a class semi-infinite programming problems in which the constraints are indexed by a compact set. We will demonstrate the usefulness of such problems by casting several important optimization problems in this form and then using semi-infinite programming techniques to solve them. Historically, Fritz John [148] initiated semi-infinite programming precisely to deduce important results about two such geometric problems: the problems of covering a compact body in $\mathbb{R}^n$ by the minimum-volume disk and the minimum-volume ellipsoid. In the same landmark paper, he derived what are now called Fritz John optimality conditions for this class of semi-infinite programs.