Calmness of efficient solution maps in parametric vector optimization

The paper is concerned with the stability theory of the efficient solution map of a parametric vector optimization problem. Utilizing the advanced tools of modern variational analysis and generalized differentiation, we study the calmness of the efficient solution map. More explicitly, new sufficient conditions in terms of the Fréchet and limiting coderivatives of parametric multifunctions for this efficient solution map to have the calmness at a given point in its graph are established by employing the approach of implicit multifunctions. Examples are also provided for analyzing and illustrating the results obtained. © 2011 Springer Science+Business Media, LLC

On the stability of the optimal value and the optimal set in optimization problems

The paper develops a stability theory for the optimal value and the optimal set mapping of optimization problems posed in a Banach space. The problems considered in this paper have an arbitrary number of inequality constraints involving lower semicontinuous (not necessarily convex) functions and one closed abstract constraint set. The considered perturbations lead to problems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the abstract constraint set can also be perturbed. The spaces of functions involved in the problems (objective and constraints) are equipped with the metric of the uniform convergence on the bounded sets, meanwhile in the space of closed sets we consider, coherently, the Attouch-Wets topology. The paper examines, in a unified way, the lower and upper semicontinuity of the optimal value function, and the closedness, lower and upper semicontinuity (in the sense of Berge) of the optimal set mapping. This paper can be seen as a second part of the stability theory presented in [17], where we studied the stability of the feasible set mapping (completed here with the analysis of the Lipschitz-like property)

Recent advances in multiobjective convex semi-infinite optimization

This paper reviews the existing literature on multiobjective (or vector) semi-infinite optimization problems, which are defined by finitely many convex objective functions of finitely many variables whose feasible sets are described by infinitely many convex constraints. The paper shows several applications of this type of optimization problems and presents a state-of-the-art review of its methods and theoretical developments (in particular, optimality, duality, and stability)

Pseudo-Lipschitz property of linear semi-infinite vector optimization problems

This paper is devoted to the study of the pseudo-Lipschitz property of Pareto solution map for the parametric linear semi-infinite vector optimization problem (LSVO). We establish new sufficient conditions for the pseudo-Lipschitz property of the Pareto solution map of (LSVO) under continuous perturbations of the right-hand side of the constraints and linear perturbations of the objective function. Examples are given to illustrate the results obtained.Linear semi-infinite vector optimization Pareto solution map Pseudo-Lipschitz mappings Linear perturbations Slater condition