Necessary Conditions in Multiobjective Optimization with Equilibrium Constraints

Abstract

Abstract. In this paper we study multiobjective optimization problems with equilibrium constraints (MOECs) described by generalized equations in the formwhere both mappings G and Q are set-valued. Such models particularly arise from certain optimization-related problems governed by variational inequalities and first-order optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the general problems under consideration and for their important specifications using modern tools of variational analysis and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical example from bilevel programming with convex while nondifferentiable data.

Variational Analysis of Marginal Functions with Applications to Bilevel Programming

“…Paper [28] proposed another approach to optimistic bilevel programs combining the aforementioned MPEC and value function ones and taking advantages of both for problems with smooth data. We also mention related developments in [11] for semi-infinite and infinite bilevel programs with DC (difference of convex) data and in [1,26] for multiobjective bilevel programs.…”

New Applications of Variational Analysis to Optimization and Control

“…However, this construction does not have a number of natural properties expected for an appropriate notion of normals. In particular, we may haveN(x; Ω ) = {0} for boundary points of Ω even in simple finite-dimensional nonconvex settings; furthermore, inevitable required calculus rules often fail for (2). The situation is dramatically improved while applying the regularization procedure…”

“…The construction (3) is known as the (basic, limiting, Mordukhovich) normal cone to Ω atx ∈ Ω ; it was introduced in [27] in an equivalent form in finite dimensions. Both constructions (2) and (3) reduce to the classical normal cone of convex analysis for convex sets Ω . In contrast to (2), the basic normal cone (3) is often nonconvex while satisfying the required properties and calculus rules in the Asplund space setting, together with the corresponding coderivative constructions for set-valued mappings and subdifferential constructions for extended-real-valued functions generated by it; see below.…”

“…In the recent papers [2,4,37,38,46,50,51] we derive necessary optimality conditions for various MPECs (22) as well as for related multiobjective optimization and equilibrium problems with equilibrium constraints governed by generalized equations/ variational conditions (23)- (27), (30)- (32) and their specifications. A major role in these conditions is played by the Fredholm constraint qualification, which reads, in the particular case of the generalized equation in (22) with a smooth base, as that the adjoint generalized equation…”

“…In [32,Sect. 5.3] and in the subsequent papers [2,37,40] we paid the main attention to the study of generalized order optimality defined as follows: given an ordering set Θ ⊂ Z with 0 ∈ Θ , we say tatx ∈ Ω is a locally ( f ,Θ , Ω )-optimal if there are a neighborhood U ofx and a sequence {z k } ⊂ Z with z k → 0 as k → ∞ such that…”