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     

生成锥内部凸-锥-类凸集值优化问题的标量化和鞍点

讨论集值向量优化的标量化和鞍点问题.在生成锥内部凸-锥-类凸假设下,建立了集值向量优化问题在（弱）有效和Benson真有效意义下的标量化定理和鞍点定理.     

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.     

序线性空间中广义半似凸集值映射向量优化的Benson真有效性

在序线性空间中建立了广义半似凸集值映射的择一定理.利用向量闭包,引进了集值优化的Benson真有效解.在广义半似凸的假设下,获得了Benson真有效性意义下的标量化定理,Lagrangian乘子定理和鞍点定理.     

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。     

非凸向量集值优化Benson真有效解的最优性条件与对偶

在无需偏序锥内部非空的情况下给出了非凸约束向量集值优化Benaon真有效解一种加细的最优性条件，并建立了向量集值优化Benson真有效解一种改进的Lagrange乘子型对偶，它比已有的Lagrange乘子型对偶具有较好的对偶性。     

关于向量集值优化的Benson真有效性

对广义锥次数凸向量集值优化问题Benson真有效性解的标量化问题进行了研究,借助于一种新的择一性定理建立了广义锥次类凸向是集值优化问题Benson真有效解的Lagrange乘子型定理并讨论了乘子型对偶问题。     

集值映射多目标半定规划的Benson真有效性

将多目标半定规划问题推广到集值映射,在广义锥-次类凸条件下,在Benson真有效性意义下研究了问题的标量化,Lagrange函数与无约束化,真鞍点条件和对偶性.     

Characterizations of the Benson Proper Efficiency for Nonconvex Vector Optimization

Under generalized cone-subconvexlikeness for vector-valued mappings in locally-convex Hausdorff topological vector spaces, a Gordan-form alternative theorem is derived. Some characterizations of the Benson proper efficiency under this generalized convexity are established in terms of scalarization, Lagrangian multipliers, saddle-point criterion, and duality.     

Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality.     

Coupled fixed point theorems in cone metric spaces

In this paper, by considering two metrics, we obtain some coupled fixed point theorems in cone metric spaces by assuming that the cone has nonempty interior as well as employing a number of contractive-type conditions. Our results generalize and extend some recently announced results in the literature.     

多目标优化问题Proximal真有效解的最优性条件

在广义凸性假设下,给出了集合proximal真有效点的线性标量化,并在此基础上证明了它与Benson真有效点和Borwein真有效点的等价性.将这些结果应用到多目标优化问题上,得到proximal真有效解的最优性条件.最后,利用proximal次微分,得到了proximal真有效解的模糊型最优性条件.     

非光滑向量均衡问题近似拟全局真有效解的最优性条件

本文研究一类非光滑向量均衡问题(Vector Equilibrium Problem)(VEP)关于近似拟全局真有效解的最优性条件.首先,利用凸集的拟相对内部型分离定理和Clarke次微分的性质,得到了问题(VEP)关于近似拟全局真有效解的最优性必要条件.其次,引入近似伪拟凸函数的概念,并给出具体实例验证其存在性,且在该凸性假设下建立了问题(VEP)关于近似拟全局真有效解的充分条件.最后,利用Tammer函数以及构建满足一定性质的非线性泛函,得到了问题(VEP)近似拟全局真有效解的标量化定理.     

Proper Efficiency in Vector Optimization on Real Linear Spaces

In this paper, we use an algebraic type of closure, which is called vector closure, and through it we introduce some adaptations to the proper efficiency in the sense of Hurwicz, Benson, and Borwein in real linear spaces without any particular topology. Scalarization, multiplier rules, and saddle-point theorems are obtained in order to characterize the proper efficiency in vector optimization with and without constraints. The usual convexlikeness concepts used in such theorems are weakened through the vector closure.     

Scalarization and lagrange duality in multiobjective optimization

In this paper, we are concerned with scalarization and the Lagrange duality in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove two theorems on scalarization and three theorems of the Lagrange duality.     

集值均衡问题近似Benson真有效解的非线性刻画

在一般的数学模型中,由于要忽略一些次要因素,所建的模型往往是近似的,且对数学模型利用数值算法所求得的解大多是近似解。另一方面,在可行集非紧的情况下,精确解的解集往往是空集,而在较弱的条件下近似解集可以是非空的。在Hausdorff局部凸拓扑线性空间中分别研究了无约束和带约束集值均衡问题近似Benson真有效解。在没有任何凸性假设下,利用非线性泛函分别建立了最优性条件。     

Infinite dimensional duality and applications

The usual duality theory cannot be applied to infinite dimensional problems because the underlying constraint set mostly has an empty interior and the constraints are possibly nonlinear. In this paper we present an infinite dimensional nonlinear duality theory obtained by using new separation theorems based on the notion of quasi-relative interior, which, in all the concrete problems considered, is nonempty. We apply this theory to solve the until now unsolved problem of finding, in the infinite dimensional case, the Lagrange multipliers associated to optimization problems or to variational inequalities. As an example, we find the Lagrange multiplier associated to a general elastic–plastic torsion problem.     

A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant

We present an interior proximal method with Bregman distance, for solving the minimization problem with quasiconvex objective function under nonnegative constraints. The Bregman function is considered separable and zone coercive, and the zone is the interior of the positive orthant. Under the assumption that the solution set is nonempty and the objective function is continuously differentiable, we establish the well definedness of the sequence generated by our algorithm and obtain two important convergence results, and show in the main one that the sequence converges to a solution point of the problem when the regularization parameters go to zero.     

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。     

A note on the paper “Fractional programming with convex quadratic forms and functions” by H.P. Benson

In this technical note, we give a short proof based on some standard results in convex analysis of some important characterization results listed in Theorems 3 and 4 of Benson [Benson, H.P., 2006. Fractional programming with convex quadratic forms and functions. European Journal of Operational Research]. Actually our result is slightly more general since we do not specify the nonempty convex set X. For clarity we use the same notation for the different equivalent optimization problems as done in Benson (2006).