A useful approximation scheme for Lagrangians

We present a new, useful approximation scheme for the integrand of an integral functional, revolving around a generalized bipolarity result. This scheme leads immediately to lower semicontinuity and lower closure results for the integral functional, as well as to other, more general seminormality properties.     

Characterisation of the weak lower semicontinuity for a type of nonlocal integral functional: The n-dimensional scalar case

In this work we are going to prove the functional J defined by

is weakly lower semicontinuous in W^1,p(Ω) if and only if W is separately convex. We assume that Ω is an open set in and W is a real-valued continuous function fulfilling standard growth and coerciveness conditions. The key to state this equivalence is a variational result established in terms of Young measures.     

Gap Continuity of Multimaps

We introduce a notion of continuity for multimaps (or set-valued maps) which is mild. It encompasses both lower semicontinuity and upper semicontinuity. We give characterizations and we consider some permanence properties. This notion can be used for various purposes. In particular, it is used for continuity properties of subdifferentials and of value functions in parametrized optimization problems. We also prove an approximate selection theorem.      

The almost chebyshev property in linear semi-infinite programming

We consider a linear semi-infinite programming problem where the index set of the constraints is compact and the constraint functions are continuous on it. The set of all continuous functions on this index set as right hand sides are the parameter set. We investigate how large various unicity sets are.We state a condition on the objective function vector and the “matrix” of the problem which characterizes when the set of a parameters with a non-unique optimal point is a set of the first Baire category in the solvability set. This is the case if and only if the unicity set is a dense subset of the solvability set. Under the same assumptions it is even true that the interior of the strong unicity set is I also dense. If the index set of the constraints contains a dense subset with the property that each point1 is a G ₈-set, then the parameters of the strong unicity set, such that the optimal point satisfies the linear independence constraint qualification, are also dense.

We apply our results to a characterization of a unique continuous selection for the optimal set I mapping and to a one-sided L ₁-approximation problem     

Multifunction Optimization Problems Involving Parameters: Necessary Optimality Conditions

We prove the Fritz John and Kuhn-Tucker necessary optimality conditions for vector optimization problems involving multifunctions and parameters under relaxed assumptions.     

Lower Semicontinuity of Kernels of Closed Convex Processes and Local Reachability of Discrete-Time Systems

Let P be a topological space and, for any p ] P, G _p : X M Y be a closed convex process between arbitrary Banach spaces X and Y . This paper shows conditions under which the map p → ker G _p : = {χ ∈ X:0 ∈ G _p (χ)} is lower pseudo-Hausdorff semicontinuous in the sense of (Optimization, 48 (2000) 17). Here, unlike (Optimization, 48 (2000) 17) we do not assume that Y is reflexive. Also, we use an approach quite different from that of (Optimization, 48 (2000) 17). As an application of the above result, we give sufficient conditions for the local reachability of a discrete-time dynamical system which is described by an arbitrary (not necessarily convex) closed set-valued map in Banach spaces.     

New nonlinear scalarization functions and applications

In this paper, new nonlinear scalarization functions, which are different from the Gerstewitz function, are introduced. Some properties of these functions are discussed, and are used to prove new results on the existence of solutions of generalized vector quasi-equilibrium problems with moving cones and the lower semicontinuity of solution mappings of parametric vector quasi-equilibrium problems. Detailed comparisons between our results and those obtained by using the Gerstewitz function (for existence theorems) and by other approaches (for the case of solution stability) are given. Illustrating examples are provided.