Some results on Lipschitz properties of the optimal values in semi-infinite programming

• Authors: F. J. Toledo
• Venue: Optimization Methods & Software - THE JOINT EUROPT-OMS CONFERENCE ON OPTIMIZATION, 4-7 JULY, 2007, PRAGUE, CZECH REPUBLIC, PART I
• Year: 2008

This paper studies certain Lipschitz properties of the optimal value function of a linear semi-infinite programming problem and its dual problem in the sense of Haar. In this setting, it is already known that the optimal value function of the so-called primal problem was Lipschitz continuous around a given stable solvable problem, when perturbations of all the coefficients are allowed. Recently, a Lipschitz constant, depending only on the nominal problem data, has been computed and, the no duality gap under certain stability conditions ensures moreover that the obtained constant still holds for the dual optimal value function. Our approach here is focused on obtaining Lipschitz constants for both primal and dual optimal value functions under weaker hypothesis of stability, which do not preclude, in all the cases, the existence of duality gap. Now, the allowed perturbations are restricted to the coefficients of the objective function of the corresponding problems.