Lipschitz modulus in convex semi-infinite optimization via d.c. functions

¹ Operations Research Center, Miguel Hernández University of Elche, 03202 Elche, Alicante, Spain. canovas@umh.es; parra@umh.es
² Postdoc researcher at Operations Research Center, Miguel Hernández University of Elche, 03202 Elche, Alicante, Spain. hantoute@ua.es
³ Department of Statistics and Operations Research, University of Alicante, 03071 Alicante, Spain. marco.antonio@ua.es

Received: 23 March 2007
Revised: 21 January 2008

Abstract

We are concerned with the Lipschitz modulus of the optimal set mapping associated with canonically perturbed convex semi-infinite optimization problems. Specifically, the paper provides a lower and an upper bound for this modulus, both of them given exclusively in terms of the problem's data. Moreover, the upper bound is shown to be the exact modulus when the number of constraints is finite. In the particular case of linear problems the upper bound (or exact modulus) adopts a notably simplified expression. Our approach is based on variational techniques applied to certain difference of convex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First] (which go back to [M.J. Cánovas, J. Global Optim. 41 (2008) 1–13] and [Ioffe, Math. Surveys 55 (2000) 501–558; Control Cybern. 32 (2003) 543–554]) constitute the starting point of the present work.