Stability of Indices in the KKT Conditions and Metric Regularity in Convex Semi-Infinite Optimization

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context, Cánovas et al. (SIAM J. Optim. 18:717–732, 2007]) introduced a sufficient condition (called ENC in the present paper) for the strong Lipschitz stability of the optimal set mapping. Now, we show that ENC also entails high stability for the minimal subsets of indices involved in the KKT conditions, yielding a nice behavior not only for the optimal set mapping, but also for its inverse. Roughly speaking, points near optimal solutions are optimal for proximal parameters. In particular, this fact leads us to a remarkable simplification of a certain expression for the (metric) regularity modulus given in Cánovas et al. (J. Glob. Optim. 41:1–13, 2008]) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, 2000]; Control Cybern. 32:543–554, 2003])), which provides a key step in further research oriented to find more computable expressions of this regularity modulus. This research was partially supported by Grants MTM2005-08572-C03 (01-02) and MTM2006-27491-E (MEC, Spain, and FEDER, E.U.), ACOMP06/117-203 and ACOMP/2007/247-292 (Generalitat Valenciana, Spain), and CIO (UMH, Spain).