A class of scalarizations of vector optimization problems is studied in order to characterize weakly efficient, efficient, and properly efficient points of a nonconvex vector problem. A parallelism is established between the different solutions of the scalarized problem and the various efficient frontiers. In particular, properly efficient points correspond to stable solutions with respect to suitable perturbations of the feasible set.

Scalarization and stability in vector optimization

Given X a Banach space and f: X â ℝ âa {+∞} a proper lower semicontinuous function which is bounded from below, the Ekeland's e-variational principle asserts the existence of a point xI in X, which we call e-extremal with respect to f, which satisfies

fu >fxI-eâu-xIââfor all u ∈ X, u â xI.

By using set convergence notions Kuratowski-Painleve, Mosco, bounded Hausdorff and their epigraphical versions we study the semi continuity properties of the mapping which to f associates e-ext f the set of such e-extremal points. The key for the geometrical understanding of such properties is to consider the equivalent Phelps extremization principle which, given a closed set D in X and a partial ordering with respect to a pointed cone, associates the set of elements of D maximal with respect to this order. Direct or potential applications are given in various fields multicriteria optimization, numerical algorithmic, calculus of variations.

Stability results for Ekeland's e-variational principle and cone extremal solutions

In this paper, we give a survey on some theoretical results in vector optimization mainly related to various notions of well-posedness, approximate solutions (or efficient points) and variational principles. We lay emphasis on papers published in the past decade.

Well-posedness in Vector Optimization

On the existence and stability of approximate solutions of perturbed vector equilibrium problems

This paper characterizes upper semicontinuity and lower semicontinuity of weakly efficient solution mapping Sw and efficient solution mapping S on the space Cm (X) of continuous m-dimension-vector-real-valued objective functions on a nonempty compact set X. As applications, some more interesting results about the properties of weakly efficient solutions and efficient solutions are obtained, such as the upper semicontinuity of S at f in Cm (X) implies the closedness and connectedness of the set S(f) of efficient solutions, any essential weakly efficient solution of f can be approximated by a sequence of efficient solutions of f(i.e., Ew (f) ⊂ S(f)), and in the sense of Baire categories "most" of vector optimization problems in Cm (X) for efficient solutions are essential, and so on.

Continuity properties of solutions of vector optimization

whenever, for every x E D such that A(x) afi(xO) for i = 1,2, . . . , n, one hash(x) = fi(x,,) for each i. Pareto optimization is also referred to as uecror, multi-objecfiue or multi-criteria optimization. Multi-objective maximization problems (0.1) have been nowadays extended to the situations where f is a mapping from X to a set Y equipped with a transitive relation 2. This relation is frequently called a preference. Consider first the image A := f(D) of the constraint D. An element y. of A is called (3_)-efficient (up to indifference or in the broad sense) if, for every y E A, y z y. implies y 5 yo; it is said to be strongly z-eficient (or efficient in the narrow sense) if, for each y E A, y 2 y. implies y = y,,. The two notions coincide for antisymmetric transitive relations. We shall be primarily concerned with efficient points in the broad sense. The set of efficient points of A (with respect to 7, ) will be denoted by max A (max Z A) and that of strongly efficient points, by max, A. An element of Y is called an efficient value of (0.1) if it is an efficient point of f(D). An element x0 of D is a solution of (0.1) if f(xo) is an efficient point of f(D). We shall use the latter S to denote the set of solutions of (0.1):

Christian Malivert

Papers

Stability of efficient sets: continuity of mobile polarities