Optimization with Multivalued Mappings

In the field of nondifferentiable nonconvex optimization, one of the most intensely investigated areas is that of optimization problems involving multivalued mappings in constraints or as the objective function. This book focuses on the tremendous development in the field that has taken place since the publication of the most recent volumes on the subject. The new topics studied include the formulation of optimality conditions using different kinds of generalized derivatives for set-valued mappings (such as, for example, the coderivative of Mordukhovich), the opening of new applications (e.g., the calibration of water supply systems), or the elaboration of new solution algorithms (e.g., smoothing methods).

The book is divided into three parts. The focus in the first part is on bilevel programming. The chapters in the second part contain investigations of mathematical programs with equilibrium constraints. The third part is on multivalued set-valued optimization. The chapters were written by outstanding experts in the areas of bilevel programming, mathematical programs with equilibrium (or complementarity) constraints (MPEC), and set-valued optimization problems.

Audience

This book is intended for researchers, graduate students and practitioners in the fields of applied mathematics, operations research, and economics.

Contenu

Bilevel Programming.- Optimality conditions for bilevel programming problems.- Path-based formulations of a bilevel toll setting problem.- Bilevel programming with convex lower level problems.- Optimality criteria for bilevel programming problems using the radial subdifferential.- On approximate mixed Nash equilibria and average marginal functions for two-stage three-players games.- Mathematical Programs with Equilibrium Constraints.- A direct proof for M-stationarity under MPEC-GCQ for mathematical programs with equilibrium constraints.- On the use of bilevel programming for solving a structural optimization problem with discrete variables.- On the control of an evolutionary equilibrium in micromagnetics.- Complementarity constraints as nonlinear equations: Theory and numerical experience.- A semi-infinite approach to design centering.- Set-Valued Optimization.- Contraction mapping fixed point algorithms for solving multivalued mixed variational inequalities.- Optimality conditions for a d.c. set-valued problem via the extremal principle.- First and second order optimality conditions in set optimization.