Stability of Solutions to Perturbed Generalized Equations

The central stability results for variational inequalities and generalized equations go back to S.M. Robinson. We make extensive use of the papers Robinson, 1979; Robinson, 1980; Robinson, 1991. Robinson, 1979 studies general stability in connection with GEs and, in particular, stability of linear GEs. The concept of strong regularity was introduced and studied in Robinson, 1980. Robinson, 1991 deals with implicit maps defined by nonsmooth equations. The original version of Theorem 5.2 comes from Robinson, 1980; further studies connected with the concept of strong regularity can be found in Dontchev and Hager, 1994 and Dontchev, 1995. The results of Mordukhovich, 1994 allow to derive efficient criteria even for the pseudo-Lipschitz behaviour of the solutions to GEs. The class of not strongly regular GEs of type (5.1) with a pseudo-Lipschitz solution map S is not large, though; cf. Dontchev and Rockafellar, 1996; Kummer, 1997. There are many further results on stability of generalized equations, above all in the context of parametric programming (e.g. Kojima, 1980; Jittorntrum, 1984).Google Scholar

Theorem 5.3 presents an equivalent definition of strong regularity for polyhedral feasible sets; it comes from Robinson, 1980 and Robinson, 1985, see also Kyparisis, 1990. In our presentation, Theorem 5.4 is a direct consequence of Theorem 5.3 and Theorem 4.6. The present proof is taken from Jiang, 1997. Proposition 5.5 is just a special case of the reduction procedure, explained in Robinson, 1980 and Lemma 5.6 goes back to Robinson, 1980. Also Theorems 5.7, 5.9 and Theorem 5.8 can be found in a slightly different form in Robinson, 1980. An alternative approach to these results, based on stability theory for LCPs, was given in Mangasarian and Shiau, 1987. Finally, Proposition 5.11 originates from Outrata, 1995.Google Scholar