Nonconvex scalarization in set optimization with set-valued maps

The aim of this work is to obtain scalar representations of set-valued optimization problems without any convexity assumption. Using a criterion of solution introduced by Kuroiwa [D. Kuroiwa, Some duality theorems of set-valued optimization with natural criteria, in: Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis, World Scientific, River Edge, NJ, 1999, pp. 221-228], which is based on ordered relations between sets, we characterize this type of solutions by means of nonlinear scalarization. The scalarizing function is a generalization of the Gerstewitz's nonconvex separation function. As applications of our results we give two existence theorems for set-valued optimization problems.     

Duality in vector optimization via augmented Lagrangian

This paper is devoted to developing augmented Lagrangian duality theory in vector optimization. By using the concepts of the supremum and infimum of a set and conjugate duality of a set-valued map on the basic of weak efficiency, we establish the interchange rules for a set-valued map, and propose an augmented Lagrangian function for a vector optimization problem with set-valued data. Under this augmented Lagrangian, weak and strong duality results are given. Then we derive sufficient conditions for penalty representations of the primal problem. The obtained results extend the corresponding theorems existing in scalar optimization.     

g-dominance: Reference point based dominance for multiobjective metaheuristics

One of the main tools for including decision maker (DM) preferences in the multiobjective optimization (MO) literature is the use of reference points and achievement scalarizing functions [A.P. Wierzbicki, The use of reference objectives in multiobjective optimization, in: G. Fandel, T. Gal (Eds.), Multiple-Criteria Decision Making Theory and Application, Springer-Verlag, New York, 1980, pp. 469–486.]. The core idea in these approaches is converting the original MO problem into a single-objective optimization problem through the use of a scalarizing function based on a reference point. As a result, a single efficient point adapted to the DM’s preferences is obtained. However, a single solution can be less interesting than an approximation of the efficient set around this area, as stated for example by Deb in [K. Deb, J. Sundar, N. Udaya Bhaskara Rao, S. Chaudhuri, Reference point based multiobjective optimization using evolutionary algorithms, International Journal of Computational Intelligence Research, 2(3) (2006) 273–286]. In this paper, we propose a variation of the concept of Pareto dominance, called g-dominance, which is based on the information included in a reference point and designed to be used with any MO evolutionary method or any MO metaheuristic. This concept will let us approximate the efficient set around the area of the most preferred point without using any scalarizing function. On the other hand, we will show how it can be easily used with any MO evolutionary method or any MO metaheuristic (just changing the dominance concept) and, to exemplify its use, we will show some results with some state-of-the-art-methods and some test problems.     

Some equivalent problems in set optimization

Given a set-valued optimization problem (P), there is more than one way of defining the solutions associated with it. Depending on the decision maker’s preference, we consider the vector criterion or the set criterion. Both criteria of solution are considered together to solve problem (P) by reducing the feasible set.     

Lagrange multipliers for set-valued optimization problems associated with coderivatives

In this paper we investigate a vector optimization problem (P) where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of (P) as solutions of a scalar optimization problem (SP) with single-valued objective and constraint functions. Then applying some classical or recent results in optimization theory to (SP) and using estimates of subdifferentials of marginal functions, we obtain optimality conditions for (P) expressed in terms of Lagrange or sequential Lagrange multipliers associated with various coderivatives of the set-valued data.     

Benson真有效意义下向量集值优化的广义Fritz-John条件

借助Clarke切锥并用上图引入了关于集值映射的Clarke切导数.借助于一种新的择一性定理建立了向量集值优化问题在弱Benson真有效意义下的广义Fritz-John最优性条件,而且证明在一种伪凸的假设下,这种最优性条件还为充分的.     

Fixed point results for set-valued contractions by altering distances in complete metric spaces

Nadler’s contraction principle has led to fixed point theory of set-valued contraction in non-linear analysis. Inspired by the results of Nadler, the fixed point theory of set-valued contraction has been further developed in different directions by many authors, in particular, by Reich, Mizoguchi–Takahashi, Feng–Liu and many others. In the present paper, the concept of generalized contractions for set-valued maps in metric spaces is introduced and the existence of fixed point for such a contraction are guaranteed by certain conditions. Our first result extends and generalizes the Nadler, Feng–Liu and Klim–Wardowski theorems and the second result is different from the Reich and Mizoguchi–Takahashi results. As a consequence, we derive some results related to fixed point of set-valued maps satisfying certain conditions of integral type.     

On optimality and duality theorems of nonlinear disjunctive fractional minmax programs

This paper is concerned with the study of optimality conditions for disjunctive fractional minmax programming problems in which the decision set can be considered as a union of a family of convex sets. Dinkelbach’s global optimization approach for finding the global maximum of the fractional programming problem is discussed. Using the Lagrangian function definition for this type of problem, the Kuhn–Tucker saddle point and stationary-point problems are established. In addition, via the concepts of Mond–Weir type duality and Schaible type duality, a general dual problem is formulated and some weak, strong and converse duality theorems are proven.     

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.     

Benson Proper Efficiency in the Vector Optimization of Set-Valued Maps

This paper extends the concept of cone subconvexlikeness of single-valued maps to set-valued maps and presents several equivalent characterizations and an alternative theorem for cone-subconvexlike set-valued maps. The concept and results are then applied to study the Benson proper efficiency for a vector optimization problem with set-valued maps in topological vector spaces. Two scalarization theorems and two Lagrange multiplier theorems are established. After introducing the new concept of proper saddle point for an appropriate set-valued Lagrange map, we use it to characterize the Benson proper efficiency. Lagrange duality theorems are also obtained     

Generalized Higher-Order Optimality Conditions for Set-Valued Optimization under Henig Efficiency

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn–Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.     

On Ha＇s Version of Set-valued Ekeland＇s Variational Principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.     

Descent methods for a class of generalized variational inequalities

In this paper we propose a class of differentiable gap functions in order to formulate a generalized variational inequality (GVI) problem, involving a set-valued map with closed and convex graph, as an optimization problem. We also show that under appropriate assumptions on the set-valued map, any stationary point of the equivalent optimization problem is a global optimal solution and solves the GVI. Finally, we describe descent methods for solving the optimization problem equivalent to the GVI and we prove its global convergence.     

Higher-Order Optimality Conditions for Set-Valued Optimization

This paper deals with higher-order optimality conditions of set-valued optimization problems. By virtue of the higher-order derivatives introduced in (Aubin and Frankowska, Set-Valued Analysis, Birkhäuser, Boston, [1990]) higher-order necessary and sufficient optimality conditions are obtained for a set-valued optimization problem whose constraint condition is determined by a fixed set. Higher-order Fritz John type necessary and sufficient optimality conditions are also obtained for a set-valued optimization problem whose constraint condition is determined by a set-valued map.     

Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure

In this paper, we define two new concepts of efficiency for vector optimization with variable ordering structure, namely the sharp and robust efficiencies, and we study their connections with classical concepts of efficiency in vector optimization. Then, we get necessary optimality conditions for them using Fréchet and Mordukhovich calculus coupled with the Gerstewitz’s (Tammer’s) scalarizing functional and openness results for set-valued maps.     

向量集值优化超有效解的对偶问题

借助于Contingent切锥和集值映射的上图而引入的有关集值映射的Contingent切导数，对约束集值优化问题的超有效解建立了最优性Kuhn Tucker必要及充分性条件,借此建立了向量集值优化超有效解的Wolfe型和Mond Weir型对偶定理.     

Experiments with classification-based scalarizing functions in interactive multiobjective optimization

In multiobjective optimization methods, the multiple conflicting objectives are typically converted into a single objective optimization problem with the help of scalarizing functions and such functions may be constructed in many ways. We compare both theoretically and numerically the performance of three classification-based scalarizing functions and pay attention to how well they obey the classification information. In particular, we devote special interest to the differences the scalarizing functions have in the computational cost of guaranteeing Pareto optimality. It turns out that scalarizing functions with or without so-called augmentation terms have significant differences in this respect. We also collect a set of mostly nonlinear benchmark test problems that we use in the numerical comparisons.     

Existence theorems for variational inclusion problems and the set-valued vector Ekeland variational principle in a complete metric space

In this paper, we apply an existence theorem for the variational inclusion problem to study the existence results for the variational intersection problems in Ekeland’s sense and the existence results for some variants of set-valued vector Ekeland variational principles in a complete metric space. Our results contain Ekeland’s variational principle as a special case and our approaches are different to those for any existence theorems for such problems.     

集值映射向量优化的Benson真有效性

本文首先将单值映射的锥次类凸概念推广到集值映射，并对锥次类凸集值映射给出几个等价刻划和一个择一性定理。然后，利用这些概念与结果来确定拓扑线性空间中带集值映射的向量优化问题的Ｂｅｎｓｏｎ真有效性，获得两个标量化结果和两个Ｌａｇｒａｎｇｅ乘子定理，在定义了一个适当的集值Ｌａｇｒａｎｇｅ映射并对其引入真鞍点的概念之后，又建立了Ｂｅｎｓｏｎ真有效性的一个充分条件和一个充要条件，最后还讨论了两个对偶问题。     

Robinson－Ursescu Theorem in p－normed Spaces

For a convex set-valued map between p-normed (0 < p ≤ 1) spaces, we give a criterion for its inverse to be locally Lipschitz of order p. From this we obtain the Robinson-Ursescu Theorem in p-normed spaces and the open mapping and closed graph theorems for closed convex set-valued maps.