非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结果;我们研究了当算子为余正仿射算子时,给出混合仿射向量变分不等式不存在例外簇的充分条件,得到混合仿射向量变分不等式弱有效解的存在性,给出了混合仿射向量变分不等式的弱有效解集为非空紧致集的充分条件.将Iusem等人(2019)在有限维空间中标量混合变分不等式解的存在性结果推广到自反Banach空间中混合向量变分不等式.     

Existence and stability of solutions for a class of generalized vector equilibrium problems

In this paper, two existence theorems concerning the strong efficient solutions and the weakly efficient solutions of generalized vector equilibrium problems are derived by using the Fan-KKM Theorem and an existence theorem for the efficient solutions of generalized vector equilibrium problems is established by using the scalarization method. Moreover, the lower semicontinuity of the strong efficient solution mapping and the weakly efficient solution mapping to parametric generalized vector equilibrium problems are showed under suitable conditions with neither monotonicity nor any information of the solution mappings. Finally, some applications to the vector optimization problems and the Stackelberg equilibrium problem are also given.     

一类集值映射向量优化问题的ε-有效解

集值映射向量优化问题是最优化理论中的一个重要方向.在集值映射为生成锥内部-锥一类凸(简记为ic-锥类凸)的假设条件下,利用择一定理,给出了集值映射向量优化问题ε-弱有效解和ε-有效解的最优性条件和ε-Lagrange乘子定理,是弱有效解和有效解相应结果的推广.     

对称向量拟均衡问题有效解的存在性

利用标量化方法建立对称向量拟均衡问题有效解的存在性定理。作为标量化方法的应用，利用这一方法得到向量变分不等式和拟向量变分不等式有效解的存在性定理。     

Generalized asymptotic functions in nonconvex multiobjective optimization problems

In this paper, we use generalized asymptotic functions and second-order asymptotic cones to develop a general existence result for the nonemptiness of the proper efficient solution set and a sufficient condition for the domination property in nonconvex multiobjective optimization problems. A new necessary condition for a point to be efficient or weakly efficient solution is given without any convexity assumption. We also provide a finer outer estimate for the asymptotic cone of the weakly efficient solution set in the quasiconvex case. Finally, we apply our results to the linear fractional multiobjective optimization problem.     

On Efficient Solutions in Vector Optimization

A new characterization is obtained for the existence of an efficient solution of a vector optimization problem in terms of associated scalar optimization problems. The consequences for linear vector optimization problems are derived as a special case, Applications to convex vector optimization problems are also discussed.     

Characterizing Nonemptiness and Compactness of the Solution Set of a Convex Vector Optimization Problem with Cone Constraints and Applications

In this paper, we characterize the nonemptiness and compactness of the set of weakly efficient solutions of a convex vector optimization problem with cone constraints in terms of the level-boundedness of the component functions of the objective on the perturbed sets of the original constraint set. This characterization is then applied to carry out the asymptotic analysis of a class of penalization methods. More specifically, under the assumption of nonemptiness and compactness of the weakly efficient solution set, we prove the existence of a path of weakly efficient solutions to the penalty problem and its convergence to a weakly efficient solution of the original problem. Furthermore, for any efficient point of the original problem, there exists a path of efficient solutions to the penalty problem whose function values (with respect to the objective function of the original problem) converge to this efficient point.     

Parametric vector optimization

In this paper we consider a sequence of vector optimization problems. We aim to generalize a vector condition that relates the parametric function and the limit function. In particular, we recover our condition given in the scalar case. Our stability approach is such that the limit of the sequence of solutions that correspond to vector optimization problems to be a solution of a limit vector optimization problem. Therefore, one can view our statement as an existence result. This general framework has been used in several previous works. In our main theorem, we use the notion of strong lower cone-semi-continuity. An example is given to illustrate why only cone-lower semi-continuity for the limit function is not sufficient for our result.     

Optimality Conditions for Vector Equilibrium Problems in Terms of Contingent Epiderivatives

In this article, we study some important properties of contingent epiderivatives concerning steady functions and a cone with a compact base along with its applications to establish necessary and sufficient optimality conditions for weakly efficient, Henig efficient, globally efficient and superefficient solutions for no constraints and constraints (it concludes cone constraint, equality constraint and a constraint set) vector equilibrium problems in terms of contingent epiderivatives. We also give some examples to illustrate obtained results.     

锥凸对称向量拟均衡问题解集的通有稳定性

在拓扑向量空间中,利用Ky Fan截口定理得到一个锥凸向量拟均衡问题弱Pareto解的存在性结果.作为该结果的应用,得到了一个对称向量拟均衡问题在支付映射为锥凸条件下弱Pareto解的存在性定理.该定理在较弱的条件下回答了Fu在文献[1]中提出的第二个问题,即在支付映射为锥凸且连续的条件下对称向量拟均衡问题的弱Pareto解是否存在.最后在赋范线性空间中研究了锥凸对称向量拟均衡问题弱Pareto解集的通有稳定性.     

On Nondifferentiable and Nonconvex Vector Optimization Problems

In this paper, we prove the equivalence among the Minty vector variational-like inequality, Stampacchia vector variational-like inequality, and a nondifferentiable and nonconvex vector optimization problem. By using a fixed-point theorem, we establish also an existence theorem for generalized weakly efficient solutions to the vector optimization problem for nondifferentiable and nonconvex functions.     

集值映射的对称向量拟均衡问题

研究向量集值映射的拟均衡问题的有效解,利用数值化方法与不动点定理,得到解的存在性定理.作为应用,得到广义向量鞍点,向量变分不等式与向量互补问题的存在性定理.     

Second-Order Optimality Conditions for Set-Valued Vector Equilibrium Problems

In this article, by using the generalized second-order contingent (adjacent) epiderivatives of set-valued maps, we obtain necessary optimality conditions and sufficient optimality conditions for weakly efficient solutions, Henig efficient solutions to the set-valued vector equilibrium problems with constraints. Some results of this article improve the corresponding results in literatures by lessening the assumption of convexity.     

带约束条件集值优化问题近似Henig有效解集的连通性

研究了带约束条件集值优化问题近似Henig有效解集的连通性.在实局部凸Hausdorff空间中,讨论了可行域为弧连通紧的,目标函数为C-弧连通的条件下,带约束条件集值优化问题近似Henig有效解集的存在性和连通性.并给出了带约束条件集值优化问题近似Henig有效解集的连通性定理.     

较多有效解类的最优性条件和Lagrange对偶定理

利用较多锥的内部和闭包,引进多目标规划问题的严格强较多有效解等概念.根据它们的表示定理,建立各类较多有效解的最优性条件,并由此得到严格弱较多有效解的Lagrange直接对偶和严格强较多有效解的Lagrange逆对偶定理.     

Contractibility of the Solution Sets in Strictly Quasiconcave Vector Maximization on Noncompact Domains

We study the contractibility of the efficient solution set of strictly quasiconcave vector maximization problems on (possibly) noncompact feasible domains. It is proved that the efficient solution set is contractible if at least one of the objective functions is strongly quasiconcave and any intersection of level sets of the objective functions is a compact (possibly empty) set. This theorem generalizes the main result of Benoist (Ref.1), which was established for problems on compact feasible domains.The authors thank Dr. T. D. Phuong, Dr. T. X. D. Ha, and the referees for helpful comments and suggestions.     

Solution concepts in vector optimization: a fresh look at an old story

Over the past decades various solution concepts for vector optimization problems have been established and used: among them are efficient, weakly efficient and properly efficient solutions. In contrast to the classical approach, we define a solution to be a set of efficient solutions on which the infimum of the objective function with respect to an appropriate complete lattice (the space of self-infimal sets) is attained. The set of weakly efficient solutions is not considered to be a solution, but weak efficiency is essential in the construction of the complete lattice. In this way, two classic concepts are involved in a common approach. Several different notions of semicontinuity are compared. Using the space of self-infimal sets, we can show that various originally different concepts coincide. A Weierstrass existence result is proved for our solution concept. A slight relaxation of the solution concept yields a relationship to properly efficient solutions.     

Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone

We define weakly minimal elements of a set with respect to a convex cone by means of the quasi-interior of the cone and characterize them via linear scalarization, generalizing the classical weakly minimal elements from the literature. Then we attach to a general vector optimization problem, a dual vector optimization problem with respect to (generalized) weakly efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem, we derive vector dual problems with respect to weakly efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.     

Semistrictly quasiconvex mappings and non-convex vector optimization

This paper introduces a new class of non-convex vector functions strictly larger than that of P-quasiconvexity, with P ^m being the underlying ordering cone, called semistrictly ( ^m\ –int P)-quasiconvex functions. This notion allows us to unify various results on existence of weakly efficient (weakly Pareto) optima. By imposing a coercivity condition we establish also the compactness of the set of weakly Pareto solutions. In addition, we provide various characterizations for the non-emptiness, convexity and compactness of the solution set for a subclass of quasiconvex vector optimization problems on the real-line. Finally, it is also introduced the notion of explicit ( ^m\ –int P)-quasiconvexity (equivalently explicit (int P)-quasiconvexity) which plays the role of explicit quasiconvexity (quasiconvexity and semistrict quasiconvexity) of real-valued functions.Acknowldegements.The author wishes to thank both referees for their careful reading of the paper, their comments, remarks, helped to improve the presentation of some results. One of the referee provided the references [5, 6] and indirectly [20].     

Non-conflicting ordering cones and vector optimization in inductive limits

Let (E, ξ)= ind (E_n, ξ_n) be an inductive limit of a sequence (E_n, ξ_n)_{n∈ N} of locally convex spaces and let every step (E_n, ξ_n) be endowed with a partial order by a pointed convex (solid) cone S_n. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence (S_n)_{n∈ N} of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.