Convex Analysis: (pms-28)

We introduce several solution concepts for multicriteria optimization problems, give a characterization of approximately efficient elements and discuss a general scalarization procedure. Furthermore, we derive necessary and sufficient optimality conditions, a minimal point theorem, a vector-valued variational principle of Ekeland’s type, Lagrangean multiplier rules and duality statements. An overview on vector variational inequalities and vector equilibria is given. Moreover, we discuss the results for special classes of vector optimization problems (vector-valued location and approximation problems, multicriteria fractional programming and optimal control problems).

Theory of Vector Optimization

In this bibliography main directions of research as well as main fields of applications of bilevel programming problems and mathematical programs with equilibrium constraints are summarized. Focus is also on the difficulties arising from nonuniqueness of lower-level optimal solutions and on optimality conditions.

Annotated Bibliography on Bilevel Programming and Mathematical Programs with Equilibrium Constraints

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Perturbation Analysis Of Optimization Problems

In this paper we introduce and study enhanced notions of relative Pareto minimizers for constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we establish the existence of relative Pareto minimizers for general multiobjective problems under a refined version of the subdifferential Palais-Smale condition for set-valued mappings with values in partially ordered spaces and then derive necessary optimality conditions for these minimizers (as well as for conventional efficient and weak efficient counterparts) that are new in both finite-dimensional and infinite-dimensional settings. Our proofs are based on variational and extremal principles of variational analysis; in particular, on new versions of the Ekeland variational principle and the subdifferential variational principle for set-valued and single-valued mappings in infinite-dimensional spaces.

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Relative Pareto minimizers for multiobjective problems: existence and optimality conditions

A concept of well-posedness, or more exactly of stability in a metric sense, is introduced for minimization problems on metric spaces generalizing the notion due to Tykhonov to situations in which there is no uniqueness of solutions. It is compared with other concepts, in particular to a variant of the notion after Hadamard reformulated via a metric semicontinuity approach. Concrete criteria of well-posedness are presented, e.g., for convex minimization problems.

Metrically well-set minimization problems

We investigate upper semicontinuity of solutions in parametric vector optimization. In particular we distinguish a class of well posed problems for which sufficient conditions for upper semicontinuity of solutions are formulated without any compactness assumptions of feasible solution sets.

Well Posedness of Vector Optimization Problems

We prove an Ekeland’s type vector variational principle for monotonically semicontinuous mappings with perturbations given by a convex bounded subset of directions multiplied by the distance function. This generalizes the existing results where directions of perturbations are singletons.

Vector variational principle

A bstract: In t he present paper we give an alternative definition of contingent epiderivative for a set-valued map. VIe usc our concept of co ntin ge nt epiderivativc to formula te necessary and/ or sufficient optimali ty conditions for a set-valued optimization problem and to study sensitivity of a family of parametrized vector optimization problems.

Ewa M. Bednarczuk

Papers

Metrically well-set minimization problems

Well Posedness of Vector Optimization Problems

Vector variational principle

Contingent epiderivative and its applications to set-valued optimization

An approach to well-posedness in vector optimization: consequences to stability