Daishi Kuroiwa

Existence theorems of set optimization with set-valued maps

In optimization with set-valued maps, there are two types of criteria of solutions; one is vector optimization, which is the usual one, and the other is set optimization, which is based on a natural idea and defined by this paper. The aim of the paper is to study set optimization. We consider criteria of solutions of set optimization, show some examples with respect to the criteria, introduce some type of semicontinuities for set-valued maps, and show existence theorems of solutions.

Convexity for set-valued maps

Abstract We establish some notions of convexity of set-valued maps. This notions are generalization of the notions of convexity of notions of convexity of vector-valued maps. Also we investigate some relations among the generalized concepts of convexity.

The relationship between multi-objective robustness concepts and set-valued optimization

In this paper, we discuss the connection between concepts of robustness for multi-objective optimization problems and set order relations. We extend some of the existing concepts to general spaces and cones using set relations. Furthermore, we derive new concepts of robustness for multi-objective optimization problems. We point out that robust multi-objective optimization can be interpreted as an application of set-valued optimization. Furthermore, we develop new algorithms for solving uncertain multi-objective optimization problems. These algorithms can be used in order to solve a special class of set-valued optimization problems.

The convexity of A and B assures intA+B=int(A+B)

Abstract The aim of this paper is to call attention to some elementary properties of convex sets for the core operator and the interior operator in a linear space and a linear topological space, respectively.

On Set Containment Characterization and Constraint Qualification for Quasiconvex Programming

Dual characterizations of the containment of a convex set with quasiconvex inequality constraints are investigated. A new Lagrange-type duality and a new closed cone constraint qualification are described, and it is shown that this constraint qualification is the weakest constraint qualification for the duality.

Surrogate duality for robust optimization

Robust optimization problems, which have uncertain data, are considered. We prove surrogate duality theorems for robust quasiconvex optimization problems and surrogate min–max duality theorems for robust convex optimization problems. We give necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, and show some examples at which such duality results are used effectively. Moreover, we obtain a surrogate duality theorem and a surrogate min–max duality theorem for semi-definite optimization problems in the face of data uncertainty.

Necessary and Sufficient Constraint Qualification for Surrogate Duality

In mathematical programming, constraint qualifications are essential elements for duality theory. Recently, necessary and sufficient constraint qualifications for Lagrange duality results have been investigated. Also, surrogate duality enables one to replace the problem by a simpler one in which the constraint function is a scalar one. However, as far as we know, a necessary and sufficient constraint qualification for surrogate duality has not been proposed yet. In this paper, we propose necessary and sufficient constraint qualifications for surrogate duality and surrogate min–max duality, which are closely related with ones for Lagrange duality.

Characterizations of the solution set for quasiconvex programming in terms of Greenberg---Pierskalla subdifferential

In convex programming, characterizations of the solution set in terms of the subdifferential have been investigated by Mangasarian. An invariance property of the subdifferential of the objective function is studied, and as a consequence, characterizations of the solution set by any solution point and any point in the relative interior of the solution set are given. In quasiconvex programming, however, characterizations of the solution set by any solution point and an invariance property of Greenberg–Pierskalla subdifferential, which is one of the well known subdifferential for quasiconvex functions, have not been studied yet as far as we know. In this paper, we study characterizations of the solution set for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. To the purpose, we show an invariance property of Greenberg–Pierskalla subdifferential, and we introduce a necessary and sufficient optimality condition by Greenberg–Pierskalla subdifferential. Also, we compare our results with previous ones. Especially, we prove some of Mangasarian’s characterizations as corollaries of our results.

Another observation on conditions assuring intA + B = int(A + B)

Abstract The aim of this paper is to give a new condition assuring cor A + B = cor( A + B ) and int A + B = int( A + B ) in a linear space and a linear topological space, respectively. Moreover, this paper presents some observations on the previous results given by the authors.

Characterizations of the solution set for non-essentially quasiconvex programming

Characterizations of the solution set in terms of subdifferentials play an important role in research of mathematical programming. Previous characterizations are based on necessary and sufficient optimality conditions and invariance properties of subdifferentials. Recently, characterizations of the solution set for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential are studied by the authors. Unfortunately, there are some examples such that these characterizations do not hold for non-essentially quasiconvex programming. As far as we know, characterizations of the solution set for non-essentially quasiconvex programming have not been studied yet. In this paper, we study characterizations of the solution set in terms of subdifferentials for non-essentially quasiconvex programming. For this purpose, we use Martínez–Legaz subdifferential which is introduced by Martínez–Legaz as a special case of c-subdifferential by Moreau. We derive necessary and sufficient optimality conditions for quasiconvex programming by means of Martínez–Legaz subdifferential, and, as a consequence, investigate characterizations of the solution set in terms of Martínez–Legaz subdifferential. In addition, we compare our results with previous ones. We show an invariance property of Greenberg–Pierskalla subdifferential as a consequence of an invariance property of Martínez–Legaz subdifferential. We give characterizations of the solution set for essentially quasiconvex programming in terms of Martínez–Legaz subdifferential.