Optimality Analysis of a Class of Semi-infinite Programming Problems

Abstract

In this paper, we consider a class of semi-infinite programming problems with a parameter. As the parameter increases, we prove that the optimal values decrease monotonically. Moreover, the limit of the sequence of optimal values exists as the parameter tends to infinity. In finding the limit, we decompose the original optimization problem into a series of subproblems. By calculating the maximum optimal values to the subproblems and applying a fixed-point theorem, we prove that the obtained maximum value is exactly the limit of the sequence of optimal values under certain conditions. As a result, the limit can be obtained efficiently by solving a series of simplified subproblems. Numerical examples are provided to verify the limit obtained by the proposed method. KeywordsSemi-infinite programming • Fixed-point theorem • Filter design • Beamformer design Mathematics Subject Classification (2000) 90C34 • 47H10 • 26B10