Solvability And Primal-dual Partitions Of The Space Of Continuous Linear Semi-infinite Optimization Problems

Abstract

Different partitions of the parameter spaceof all linear semi-infinite programming problems witha fixed compact set of indices and continuous rightand left hand side coefficients have been consideredin this paper. The optimization problems are classifiedin a different manner, e.g., consistent and inconsistent, solvable (with bounded optimal value and nonempty optimal set), unsolvable (with bounded optimal valueand empty optimal set) and unbounded (with infinite optimal value). The classification we propose generatesa partition of the parameter space, called second general primal-dual partition. We characterize each cell of the partition by means of necessary and sufficient, and in some cases only necessary or sufficient conditions, assuring that the pair of problems (primal and dual) belongs to that cell. In addition, we show non emptiness of each cell of the partition and with plenty of examples we demonstrate that some of the conditions are only necessary or sufficient. Finally, we investigate various questions of stability of the presented partition.