非光滑非凸向量极值问题的真有效解

本文考虑非光滑非凸向量极值问题的真有效解,其主要结果如下:(1)Borwein真有效解与Benson真有效解的等价性;(2)向量极值问题的真有效解与标量极值问题的最优解的等价性;(3)广义鞍点定理;(4)真有效解的必要和充分条件。     

集值优化问题Borwein真有效解与Benson真有效解的等价性

在局部凸空间中锥弱似凸集值映射的假设下,集值优化问题Borwein真有效解与Benson真有效解的等价性被获得.为了说明结果,一些例子被给出.     

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.     

非光滑向量极值问题的真有效解与最优性条件

讨论了赋范线性空间中非光滑向量极值问题的Hatley,Borwein,Benson真有效解之间的关系,指出了它们共同的标量极值问题的等价刻画,建立了问题（VMP）的广义KT-真有效解的充分条件,并给出了向量极小值问题在锥局部凸、拟凸、伪凸等条件下的最优性条件。     

赋范线性空间多目标规划的 广义KT-真有效解

对于赋范线性空间中的多目标规划问题,引进了广义KT-真有效解的概念.在一定条件下,得到了广义KT-真有效解和广义H-局部真有效解之间的关系.     

多目标优化问题鲁棒有效解与真有效解之间的关系

在一定条件下研究了多目标优化问题鲁棒有效解与真有效解之间的关系及鲁棒有效解的最优性条件.首先,给出多目标优化问题鲁棒弱有效解的概念,研究它与鲁棒有效解和真有效解之间的关系,举例说明了相关结果的合理性.其次,在次类凸和伪凸性假设下研究了鲁棒有效解的必要性条件和充分性条件.     

多目标优化问题的完全标量化

本文首先利用松弛变量和广义Tchebycheff范数的推广形式提出一类新的标量化优化问题.进一步,通过调整几种参数范围获得一般多目标优化问题弱有效解、有效解和真有效解的一些完全标量化刻画.此外,本文提出例子对主要结果进行说明,利用相应的标量化方法判定给定的多目标优化问题的可行解是否是弱有效解、有效解和真有效解.     

集优化问题的近似Benson真有效解及非线性刻画

利用集合的Minkowski差,引进集优化问题的Benson真有效解和近似Benson真有效解的概念,讨论了它们之间的关系.在某种假设下证明了近似Benson真有效解集是闭的.利用非线性泛函分别建立了无约束和带约束集优化问题取得近似Benson真有效解的必要条件.     

不确定多目标优化鲁棒真有效解的最优性与对偶

本文研究一类不确定性多目标优化问题鲁棒真有效解的最优性条件和对偶理论.首先,借助鲁棒真有效解的标量化定理,在鲁棒型闭凸锥约束品性下,建立了不确定多目标优化问题真有效解的最优性条件;其次,针对原不确定多目优化的Wolfe型对偶问题,得到关于鲁棒真有效解的强、弱对偶定理.     

广义弧式连通凸锥优化问题的最优性条件及对偶问题

给出了弧式连通凸锥优化问题的强有效解和Benson真有效解的最优性条件,讨论了目标函数和约束函数均为广义弧式连通凸锥函数优化问题的近似有效解的最优性条件,给出了相应的近似Mond-Weir型对偶模型,给出了弱对偶和逆对偶定理.     

多目标规划的整体解

本文在较弱的广义凸性假定下讨论多目标规划的几种整体有效性．给出的定理统一了目前已有的一些关于多目标规划局部解为整体解的充分条件．     

拟凸函数的近似次微分及其在多目标优化问题中的应用

针对拟凸函数提出一类新的近似次微分,研究其性质,并将近似次微分应用到拟凸多目标优化问题近似解的刻画中.首先,对已有的近似次微分进行改进,得到拟凸函数新的近似次微分,并给出其与已有次微分之间的关系及一系列性质.随后,利用新的近似次微分给出拟凸多目标优化问题近似有效解、近似真有效解的最优性条件.     

向量优化问题的几个收敛性结果

在文本中，获得了集合的弱有效元与真有效元的几个收敛性结果。然后，讨论了集值映射向量优化问题（VP）和它的近似问题（VP）n，在较强的假设条件下，获得了（VP）n的真有效解的几个收敛性结果。     

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.     

An efficient gradient method with approximate optimal stepsize for the strictly convex quadratic minimization problem

In this paper, a new type of stepsize, approximate optimal stepsize, for gradient method is introduced to interpret the Barzilai–Borwein (BB) method, and an efficient gradient method with an approximate optimal stepsize for the strictly convex quadratic minimization problem is presented. Based on a multi-step quasi-Newton condition, we construct a new quadratic approximation model to generate an approximate optimal stepsize. We then use the two well-known BB stepsizes to truncate it for improving numerical effects and treat the resulted approximate optimal stepsize as the new stepsize for gradient method. We establish the global convergence and R-linear convergence of the proposed method. Numerical results show that the proposed method outperforms some well-known gradient methods.     

An approximate strong KKT condition for multiobjective optimization

In this paper, we introduce a sequential approximate strong Karush–Kuhn–Tucker (ASKKT) condition for a multiobjective optimization problem with inequality constraints. We show that each local efficient solution satisfies the ASKKT condition, but weakly efficient solutions may not satisfy it. Subsequently, we use a so-called cone-continuity regularity (CCR) condition to guarantee that the limit of an ASKKT sequence converges to an SKKT point. Finally, under the appropriate assumptions, we show that the ASKKT condition is also a sufficient condition of properly efficient points for convex multiobjective optimization problems.     

Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization

For a given multiobjective optimization problem, we study recession properties of the sets of efficient solutions and properly efficient solutions. We work out various consequences based on the obtained recession properties, including a characterization for the boundedness and nonemptiness of the set of (properly) efficient solutions when the problem is a convex problem. We also show that the boundedness and nonemptiness of the set of efficient solutions is equivalent to that of the set of properly efficient solutions under an additional mild condition. Finally, we provide some new verifiable necessary conditions for the nonemptiness of the set of efficient solutions in terms of the associated recession functions and recession cones.     

Weakly and properly nonessential objectives in multiobjective optimization problems

In this work we characterize objective functions which do not change the set of efficient solutions (weakly efficient solutions, properly efficient solutions). Necessary and sufficient conditions for an objective function to be weakly nonessential (properly nonessential) are presented. We establish relations between weakly nonessential, properly nonessential and nonessential functions.