Disjunctive Quasiconvex Programming and Other Nonconvex Problems

A disjunctive programming problem corresponds to the minimization of an objective function on a constraint set which is a union of sets, for example, min f (x) s.t. min j∈ J g j (x) ≤ 0 where the functions f and g j are real valued defined on a Banach space X and J is a (possibly infinite) index set. This class of problems has been introduced by Balas [4] in 1974 and then studied by many authors, essentially in the case where the functions f and g j are convex and for finite index set J. A general duality theory was developed in Borwein [5]. For numerical methods dedicaced to disjunctive programming, see the review of Grossmann [7] and a recent paper of Cornuejols-Lemaréchal [6]. Mathematical programming with equilibrium constraints (MPEC in short) is another class of minimization problems which naturally generates nonconvex constraint sets. Our aim is to present existence results ([2, 3]), in infinite dimensions, for quasicon-vex disjunctive problems (functions f and g j are assumed to be quasiconvex) and quasiconvex MPEC problems. Quasiconvex bilivel problems will be also considered. This study is based on the so-called normal approach of quasiconvex analysis which consists, roughly speaking, to consider the normal cone to sublevel sets. The resulting associated set-valued map, the normal operator, introduced in [1], enjoys many interesting properties.