Pareto Solutions of Polyhedral-valued Vector Optimization Problems in Banach Spaces

In general Banach spaces, we consider a vector optimization problem (SVOP) in which the objective is a set-valued mapping whose graph is the union of finitely many polyhedra. We establish some results on structure and connectedness of the weak Pareto solution set, Pareto solution set, weak Pareto optimal value set and Pareto optimal value set of (SVOP). In particular, we improve and generalize Arrow, Barankin and Blackwell’s classical results on linear vector optimization problems in Euclidean spaces.     

Fuzzy Optimization Problems Based on Ordering Cones

The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients.     

A Proximal-Type Method for Convex Vector Optimization Problem in Banach Spaces

In this paper, we consider a convex vector optimization problem of finding weak Pareto optimal solutions from a uniformly convex and uniformly smooth Banach space to a real Banach space, with respect to the partial order induced by a closed, convex, and pointed cone with a nonempty interior. We introduce a vector-valued proximal-type method based on the Lyapunov functional, carry out convergent analysis on this method, and prove that any sequence generated by the method weakly converges to a weak Pareto optimal solution of the vector optimization problem under some mild conditions. Our results improve and generalize some known results.     

Equivalence and existence of weak Pareto optima for multiobjective optimization problems with cone constraints

In this work, a differentiable multiobjective optimization problem with generalized cone constraints is considered, and the equivalence of weak Pareto solutions for the problem and for its η-approximated problem is established under suitable conditions. Two existence theorems for weak Pareto solutions for this kind of multiobjective optimization problem are proved by using a Karush–Kuhn–Tucker type optimality condition and the F-KKM theorem.     

Characterizations of solution sets for parametric multiobjective optimization problems

In this article, we investigate the nonemptiness and compactness of the weak Pareto optimal solution set of a multiobjective optimization problem with functional constraints via asymptotic analysis. We then employ the obtained results to derive the necessary and sufficient conditions of the weak Pareto optimal solution set of a parametric multiobjective optimization problem. Our results improve and generalize some known results.     

A modified wolfe dual for weak vector minimization

A version of the Wolfe dual problem is constructed for constained weak minimization of a vector objective function, in finite or infinite dimensions (e.g. continuous programming) The usual convex requirements are weakened to invex. Weak duality is replaced by an inclusion, constructed using the cone defining the weak minimum. Relations with Pareto (or proper Pareto) minima are discussed.     

A Proximal Point-Type Method for Multicriteria Optimization

In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.     

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.     

Lagrange Multiplier Rules for Weak Approximate Pareto Solutions of Constrained Vector Optimization Problems in Hilbert Spaces

In the Hilbert space case, in terms of proximal normal cone and proximal coderivative, we establish a Lagrange multiplier rule for weak approximate Pareto solutions of constrained vector optimization problems. In this case, our Lagrange multiplier rule improves the main result on vector optimization in Zheng and Ng (SIAM J. Optim. 21: 886–911, 2011). We also introduce a notion of a fuzzy proximal Lagrange point and prove that each Pareto (or weak Pareto) solution is a fuzzy proximal Lagrange point.     

双目标规划问题的像集与求解

经济管理的决策目标往往与成本、收益相关,双目标规划在经济管理中具有广泛应用.然而,尚缺乏成熟的算法确定双目标规划问题的全部解.给出双目标规划问题像集的一般性确定法,以求其解,为研究目的所在.具体而言,构造一个带等式约束的单目标规划问题,以确定双目标规划问题像集之部分边界,并借助拉格朗日乘子符号判断其单调性,据此确定原问题的帕累托解与弱帕累托解.这相当于提供了一个求解双目标规划问题的一般性框架.     

Convergence for vector optimization problems with variable ordering structure

In this paper, we first discuss the Painlevé–Kuratowski set convergence of (weak) minimal point set for a convex set, when the set and the ordering cone are both perturbed. Next, we consider a convex vector optimization problem, and take into account perturbations with respect to the feasible set, the objective function and the ordering cone. For this problem, by assuming that the data of the approximate problems converge to the data of the original problem in the sense of Painlevé–Kuratowski convergence and continuous convergence, we establish the Painlevé–Kuratowski set convergence of (weak) minimal point and (weak) efficient point sets of the approximate problems to the corresponding ones of original problem. We also compare our main theorems with existing results related to the same topic.     

关于Pareto极值问题

[1]中研究了Pareto极值问题.[3]中部分地改进了[2]中的结果.本文给出Pareto极值存在的一个一般结论,这一结论包含了[2][3]中的有关结论作为特殊情况.利用弱拓扑还讨论了强-Pareto极值的存在性,从而在比[2][3]有关定理更弱的条件下,获得了更强的结论.     

有效点、弱有效点和真有效点的锥刻画

利用集合在某点的相依切锥、法向锥和可行方向锥等研究向量优化问题的有效点、 弱有效点和真有效点的特征,对局部有效点、局部弱有效点和局部真有效点与集合的各 锥之间的关系作了刻画.     

UNIFORM APPROXIMATION OF A MULTIOBJECTIVE OPTIMIZATION SEQUENCE

In this paper the Pareto efficiency of a uniformly convergent multiobjective optimization sequence is studied. We obtain some relation between the Pareto efficient solutions of a given multiobjective optimization problem and those of its uniformly convergent optimization sequence and also some relation between the weak Pareto efficient solutions of the same optimization problem and those of its uniformly convergent optimization sequence. Besides, under a compact convex assumption for constraints set and a certain convex assumption for both objective and constraint functions, we also get some sufficient and necessary conditions that the limit of solutions of a uniformly convergent multiobjective optimization sequence is the solution of a given multiobjective optimization problem.     

Duality and a Characterization of Pseudoinvexity for Pareto and Weak Pareto Solutions in Nondifferentiable Multiobjective Programming

In this paper, we unify recent optimality results under directional derivatives by the introduction of new pseudoinvex classes of functions, in relation to the study of Pareto and weak Pareto solutions for nondifferentiable multiobjective programming problems. We prove that in order for feasible solutions satisfying Fritz John conditions to be Pareto or weak Pareto solutions, it is necessary and sufficient that the nondifferentiable multiobjective problem functions belong to these classes of functions, which is illustrated by an example. We also study the dual problem and establish weak, strong, and converse duality results.     

Pareto efficiency in locally convex spaces II: Stability results

In this second part of our work on Pareto efficiency we study the stability of the set of efficient points under perturbations of the data. For that purpose we use a general notion of convergence of sets which is known as the Kuratowski-Mosco convergence. We also determine the properties of Eff_K(S) as a function of the set S and of the dominance cone K. Finally our work is extended to stochastic Pareto efficiency and we derive several stability results, we determine the measurability of the efficiency multifunction and we obtain a new characterization of the efficient points of the aggregate set.     

Pointwise well-posedness in set optimization with cone proper sets

This paper deals with the well-posedness property in the setting of set optimization problems. By using a notion of well-posed set optimization problem due to Zhang et al. (2009) [18] and a scalarization process, we characterize this property through the well-posedness, in the Tykhonov sense, of a family of scalar optimization problems and we show that certain quasiconvex set optimization problems are well-posed. Our approach is based just on a weak boundedness assumption, called cone properness, that is unavoidable to obtain a meaningful set optimization problem.     

Generalized viscosity approximation methods in multiobjective optimization problems

In this paper, we consider an extend-valued nonsmooth multiobjective optimization problem of finding weak Pareto optimal solutions. We propose a class of vector-valued generalized viscosity approximation method for solving the problem. Under some conditions, we prove that any sequence generated by this method converges to a weak Pareto optimal solution of the multiobjective optimization problem.     

同伦方法求解非凸区域Brouwer不动点问题

本文构造了一个新的求解非凸区域上不动点问题的内点同伦算法,并在弱法锥(见定义2．1(2))和适当的条件下,证明了算法的全局收敛性．本文所给的条件比外法锥条件更加一般．