Stability Properties for Parametric Linear Programs under Data Ambiguities

Abstract

In this paper, we study a new parametric robust linear problem (PRLP) whose data are allowed to be perturbed not only on the objective and constraint functions but also on the size of the uncertainty sets. Using a dual approach, we examine the stability and sensitivity properties of PRLP by looking at how the behaviors of its optimal value function and solution map change according to the change of the parameters. More precisely, we examine the closedness and lower and upper semicontinuity of the solution map and the lower and upper semicontinuity as well as Lipschitz property of the optimal value function of PRLP varying around a reference parameter. In this way, we obtain the nonemptiness and boundedness of the solution sets and a characterization for the Lipschitz continuity of the optimal value function for semi-infinite linear programs when fixing the corresponding index sets.