Stability of Indices in the KKT Conditions and Metric Regularity in Convex Semi-Infinite Optimization
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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, ) 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, ) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, ; Control Cybern. 32:543–554, )), which provides a key step in further research oriented to find more computable expressions of this regularity modulus.
KeywordsConvex semi-infinite programming KKT conditions Modulus of metric regularity
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- 6.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation I. Springer, Berlin (2006) Google Scholar
- 7.Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1997) Google Scholar
- 11.Nürnberger, G., Unicity in semi-infinite optimization. In: Brosowski, B., Deutsch, F. (eds.) Parametric Optimization and Approximation, pp. 231–247. Birkhäuser, Basel (1984) Google Scholar
- 14.Cánovas, M.J., Gómez-Senent, F.J, Parra, J.: On the Lipschitz modulus of the argmin mapping in linear semi-infinite optimization. Set-Valued Anal. Published online 20 September 2007 Google Scholar