A note on the compactness of the index set in convex optimization. Application to metric regularity
M. J. Cánovas, Abderrahim Hantoute, Marco A. López, Juan Parra · Optimization · 2008
We extend some recent developments on metric regularity in convex semi-infinite optimization from the case of a compact metric index set (for the constraint system) to the case of a compact Hausdorff one. The latter is the continuous setting where different contributions on stability in semi-infinite programming were developed since the 1980s (see for instance 1 Brosowski, B. 1984. Parametric semi-infinite linear programming I. Continuity of the feasible set and of the optimal value. Math. Program. Study, 21: 18–42. [Crossref], [Web of Science ®] , [Google Scholar], 7 Fischer, T. 1983. “Contributions to semi-infinite linear optimization”. In Approximation and Optimization in Mathematical Physics, Edited by: Brosowski, B and Martensen, E. 175–199. Frankfurt-Am-Main: Peter Lang. [Google Scholar] and 8 Goberna, MA and López, MA. 1998. Linear Semi-Infinite Optimization, Chichester, , UK: John Wiley & Sons. [Google Scholar]). In contrast, 3 Cánovas, MJ. Stability of indices in KKT conditions and metric regularity in convex semi-infinite optimization. J. Optim. Theory Appl., (to appear) [Google Scholar] and 4 Cánovas, MJ. 2007. Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim., 18: 717–732. [Crossref], [Web of Science ®] , [Google Scholar] (see also 2 Cánovas, MJ, Gómez-Senent, FJ and Parra, J. 2006. On the Lipschitz modulus of the argmin mapping in linear semi-infinite optimization. Set-Valued Anal., publised online: September 20, 2007 [Google Scholar] for the linear case) deal with a compact metric index set when analysing the stable behaviour of KKT conditions. The fact of removing the ‘metric’ assumption yields some technical difficulties which are tackled in the present article via basic topological tools.