Optimality Conditions and a Solution Method
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In this chapter we apply the preceding theory to establish first-order necessary optimality conditions and to construct an efficient and robust numerical method for the solution of the considered MPECs. This numerical method will extensively be used in the second (“applied”) part of the book.
KeywordsAdjoint Equation Bilevel Program Equilibrium Constraint Bundle Method Sufficient Optimality Condition
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- Optimality conditions of the type developed in Section 7.1 appeared, to our knowledge, first in Outrata, 1993 in the context of bilevel programming. In Outrata, 1994 they were extended to MPECs with variational inequalities and in Kočvara and Outrata, 1997 to MPECs with implicit complementarity problems. On the basis of the results from Outrata, 1995 one could extend them also to equilibria described by quasi-variational inequalities, but the necessary requirements are then in most cases too severe.Google Scholar
- The numerical method investigated in Section 7.2 was proposed in Outrata, 1990 in the context of simple bilevel programs, where the lower-level constraints do not depend on the upper-level variable. In the subsequent papers Outrata, 1993; Outrata, 1994; Kočvara and Outrata, 1994b and Outrata and Zowe, 1995b this technique was developed to the level presented here, and applied to various types of equilibria (lower-level convex programs, variational inequalities and implicit complementarity problems).Google Scholar
- In the papers Dempe, 1995; Dempe, 1998 this method is further extended to bilevel programs with nonunique KKT vectors in the lower-level program. Particularly, (ELICQ) is replaced by some other conditions guaranteeing that S1 is locally Lipschitz and directionally differentiable (cf. also Ralph and Dempe, 1995; Pang and Ralph, 1996). Then, however, the computation of subgradients of θ becomes more complicated. In Dempe and Schmidt, 1996; Outrata, 1997 and Dempe, 1998 this method is adapted even to the solution of bilevel programs with nonunique lower-level solutions. Whereas in Outrata, 1997 only a special case motivated by network design problems is considered, the main idea of Dempe and Schmidt, 1996 consists in a suitable regularization of lower-level programs so that the “regularized” problems fulfil the standard assumptions. By using the results from Kalashnikov and Kalashnikova, 1996 this approach could be used also in MPECs with monotone (but not strongly monotone) variational inequalities. Both these generalizations deserve a thorough numerical testing.Google Scholar