Parametric semi-infinite linear programming I. continuity of the feasible set and of the optimal value

Abstract

The purpose of this paper is to study the recent development of parametric semi-infinite linear optimization. Especially, we will investigate continuity properties of the feasible set mapping and of the optimal value. Further, we will investigate the structure of certain parameter sets. We begin by stating some definitions and by recalling a necessary and sufficient condition that a set of feasible points is optimal. In Section 2, we derive various conditions for the continuity of the feasible set mapping Z. In particular, we derive a result of Fischer [16] that Z is upper semicontinuous at a parameter σ₀ if and only if $Z_{\sigma _0 }$ is compact. In Section 3, we investigate the continuity of the optimal value. We derive various conditions for the continuity of E, e.g. E is locally Lipschitz at σ₀ if $P_{\sigma _0 }$ is compact and $Z_{\sigma _0 }$ satisfies the Slater-condition. Finally, we show that certain subsets of the parameter space are convex. These results imply that E is piecewise linear and convex (resp. concave) on certain subsets of the parameter space.

Some of the following results are special cases of known more general theorems. To make the paper self-contained, we have also given proofs in these cases. We preferred to give more elementary direct proofs using the linear structure instead of deriving them from general theorems.