Multiobjective optimization problems with modified objective functions and cone constraints and applications

In this paper, we consider a differentiable multiobjective optimization problem with generalized cone constraints (for short, MOP). We investigate the relationship between weakly efficient solutions for (MOP) and for the multiobjective optimization problem with the modified objective function and cone constraints [for short, (MOP) _η (x)] and saddle points for the Lagrange function of (MOP) _η (x) involving cone invex functions under some suitable assumptions. We also prove the existence of weakly efficient solutions for (MOP) and saddle points for Lagrange function of (MOP) _η (x) by using the Karush-Kuhn-Tucker type optimality conditions under generalized convexity functions. As an application, we investigate a multiobjective fractional programming problem by using the modified objective function method.     

Some properties of approximate solutions for vector optimization problem with set-valued functions

In this paper, we study the approximate solutions for vector optimization problem with set-valued functions. The scalar characterization is derived without imposing any convexity assumption on the objective functions. The relationships between approximate solutions and weak efficient solutions are discussed. In particular, we prove the connectedness of the set of approximate solutions under the condition that the objective functions are quasiconvex set-valued functions.     

On optimization problems with set-valued objective maps

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets in n-dimensional Euclidean space and investigate their properties. Next, by using these orderings, we define the concepts of the convexities to set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and characterize their properties.     

Regularity conditions characterizing Fenchel–Lagrange duality and Farkas-type results in DC infinite programming

In this paper, we consider a DC infinite programming problem (P) with inequality constraints. By using the properties of the epigraph of the conjugate functions, we introduce some new notions of regularity conditions for (P). Under these new regularity conditions, we completely characterize the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for (P). Similarly, we also completely characterize the Farkas-type results and the stable Farkas-type results for (P). As applications, we obtain the corresponding results for conic programming problems.     

Approximate duality for vector quasi-equilibrium problems and applications

In this paper, we introduce new versions of ?-dual problems of a vector quasi-equilibrium problem with set-valued maps, and we give an ?-duality result between approximate solutions of the primal and dual problems. As the first application of the main result, we obtain an ?-duality for a vector quasi-equilibrium problem whose ?-solutions are understood in the sense of proper efficiency. The second application is devoted to an?-duality for a vector optimization problem with set-valued maps.     

二层多目标最优化问题解集的连通性

研究了二层多目标最优化模型(BLMOP)解集的连通性问题,其中(BLMOP)的上层集值目标函数由下层问题的有效点确定.把(BLMOP)看作成单层的集值函数优化问题,借助集值函数优化问题各种有效解集的连通性的结论,得到了(BLMOP)相应的有效解集连通性的结论.     

Existence results and gap functions for the generalized equilibrium problem with composed functions

In this paper we provide first existence results for solutions of the generalized equilibrium problem with composed functions (GEPC) under generalized convexity assumptions. Then we construct by employing some tools specific to the theory of conjugate duality two gap functions for (GEPC). The importance of these gap functions is to be seen in the fact that they equivalently characterize the solutions of an equilibrium problem. We also prove that for some particular instances of (GEPC) the gap functions we introduce here become among others the celebrated Auslender’s and Giannessi’s gap functions.     

拟不变凸集值优化问题严有效解的最优性条件

在赋范线性空间中借助切导数研究集值优化问题的严有效性．当目标函数和约束函数相对于同一向量函数为拟不变凸时,利用凸集分离定理给出了集值优化问题取得严有效元的Kuhn—Xhcker型最优陛必要条件．利用切导数的性质,用构造性方法得到了拟不变凸集值优化问题取得严有效元的充分条件．     

New Order Relations in Set Optimization

In this paper we study a set optimization problem (SOP), i.e. we minimize a set-valued objective map F, which takes values on a real linear space Y equipped with a pre-order induced by a convex cone K. We introduce new order relations on the power set P(Y)\mathcal{P}(Y) of Y (or on a subset of it), which are more suitable from a practical point of view than the often used minimizers in set optimization. Next, we propose a simple two-steps unifying approach to studying (SOP) w.r.t. various order relations. Firstly, we extend in a unified scheme some basic concepts of vector optimization, which are defined on the space Y up to an arbitrary nonempty pre-ordered set (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) without any topological or linear structure. Namely, we define the following concepts w.r.t. the pre-order \preccurlyeq\preccurlyeq: minimal elements, semicompactness, completeness, domination property of a subset of Q\mathcal{Q}, and semicontinuity of a set-valued map with values in Q\mathcal{Q} in a topological setting. Secondly, we establish existence results for optimal solutions of (SOP), when F takes values on (Q,\preccurlyeq)(\mathcal{Q},\preccurlyeq) from which one can easily derive similar results for the case, when F takes values on P(Y)\mathcal{P}(Y) equipped with various order relations.     

Solving Ill-posed Bilevel Programs

This paper deals with ill-posed bilevel programs, i.e., problems admitting multiple lower-level solutions for some upper-level parameters. Many publications have been devoted to the standard optimistic case of this problem, where the difficulty is essentially moved from the objective function to the feasible set. This new problem is simpler but there is no guaranty to obtain local optimal solutions for the original optimistic problem by this process. Considering the intrinsic non-convexity of bilevel programs, computing local optimal solutions is the best one can hope to get in most cases. To achieve this goal, we start by establishing an equivalence between the original optimistic problem and a certain set-valued optimization problem. Next, we develop optimality conditions for the latter problem and show that they generalize all the results currently known in the literature on optimistic bilevel optimization. Our approach is then extended to multiobjective bilevel optimization, and completely new results are derived for problems with vector-valued upper- and lower-level objective functions. Numerical implementations of the results of this paper are provided on some examples, in order to demonstrate how the original optimistic problem can be solved in practice, by means of a special set-valued optimization problem.     

On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets of n-dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi-continuity for set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.     

A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond-Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond-Weir duals, a modified Mond-Weir vector dual problem with a modified objective function is constructed.     

Four types of nonlinear scalarizations and some applications in set optimization

There are two types of criteria of solutions for the set-valued optimization problem, the vectorial criterion and set optimization criterion. The first criterion consists of looking for efficient points of set valued map and is called set-valued vector optimization problem. On the other hand, Kuroiwa–Tanaka–Ha started developing a new approach to set-valued optimization which is based on comparison among values of the set-valued map. In this paper, we treat the second type criterion and call set optimization problem. The aim of this paper is to investigate four types of nonlinear scalarizing functions for set valued maps and their relationships. These scalarizing functions are generalization of Tammer–Weidner’s scalarizing functions for vectors. As applications of the scalarizing functions for sets, we present nonconvex separation type theorems, Gordan’s type alternative theorems for set-valued map, optimality conditions for set optimization problem and Takahashi’s minimization theorems for set-valued map.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

A Hausdorff-type distance,a directional derivative of a set-valued map and applications in set optimization

Abstract

In this paper, we follow Kuroiwa’s set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F.     

Super efficiency in vector optimization with nearly convexlike set-valued maps

In this paper, we establish a scalarization theorem and a Lagrange multiplier theorem for super efficiency in vector optimization problem involving nearly convexlike set-valued maps. A dual is proposed and duality results are obtained in terms of super efficient solutions. A new type of saddle point, called super saddle point, of an appropriate set-valued Lagrangian map is introduced and is used to characterize super efficiency.     

生成锥内部凸-锥-类凸集值优化问题的Henig真有效性

该文讨论局部凸空间中的约束集值优化问题. 首先, 在生成锥内部凸-锥-类凸假设下, 建立了Henig真有效解在标量化和Lagrange乘子意义下的最优性条件. 其次, 对集值Lagrange映射引入Henig真鞍点的概念, 并用这一概念刻画了Henig真有效解. 最后, 引入了一个标量Lagrange对偶模型, 并得到了关于Henig真有效解的对偶定理. 另外, 该文所得结果均不需要约束序锥有非空的内部.     

A set-valued Lagrange theorem based on a process for convex vector programming

ABSTRACT

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.     

Hadamard well-posedness for a set-valued optimization problem

In this paper, we introduce a kind of Hadamard well-posedness for a set-valued optimization problem. By virtue of a scalarization function, we obtain some relationships between weak ${(\varepsilon, e)}$ -minimizers of the set-valued optimization problem and ${\varepsilon}$ -approximate solutions of a scalar optimization problem. Then, we establish a scalarization theorem of P.K. convergence for sequences of set-valued mappings. Based on these results, we also derive a sufficient condition of Hadamard well-posedness for the set-valued optimization problem.