近似次微分刻画的约束向量均衡问题的最优性条件北大核心CSCD

该文研究一类约束向量均衡问题(CVEP)近似拟弱有效解的最优性条件和对偶定理.首先,建立了问题(CVEP)近似拟弱有效解关于近似次微分形式的最优性必要条件.其次,引入了一种广义凸性的概念,称之为近似伪拟type-I函数,并在其假设下,获得了问题(CVEP)近似拟弱有效解的最优性充分条件.最后,引入了问题(CVEP)的广义近似Mond-Weir对偶模型,并建立其与原问题间关于近似拟弱有效解的对偶定理.     

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。     

鲁棒凸优化问题拟近似解的刻划

本文研究鲁棒凸优化问题拟近似解的最优性条件和对偶理论.首先利用鲁棒优化方法,在由约束函数的共轭函数的上图给出的闭凸锥约束规格条件下,建立了拟近似解的最优性充要条件.其次给出了鲁棒凸优化问题拟近似解在Wolf型和Mond-weir型对偶模型下的强(弱)对偶定理.最后给出具体实例验证了本文获得的结果.     

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.     

半凸多目标优化ε-拟弱有效解的最优性条件及对偶定理

本文在相关文献考虑MP问题的基础上,增加了等式约束条件,即本文考虑了VP问题,并将已有文献中的凸性假设改为半凸性假设,得到VP问题的ε-拟弱有效解的相应最优性条件.接着,本文定义了VP问题的拉格朗日函数及其ε-拟弱鞍点,得到VP问题的ε-拟弱鞍点相应定理.最后,本文考虑了VP问题的对偶问题,获得了VP问题的弱对偶和强对偶定理.     

向量映射的鞍点和Lagrange对偶问题

本文研究拓扑向量空间广义锥-次类凸映射向量优化问题的鞍点最优性条件和Lagrange对偶问题,建立向量优化问题的Fritz John鞍点和Kuhn-Tucker鞍点的最优性条件及其与向量优化问题的有效解和弱有效解之间的联系。通过对偶问题和向量优化问题的标量化刻画各解之间的关系,给出目标映射是广义锥-次类凸的向量优化问题在其约束映射满足广义Slater约束规格的条件下的对偶定理。     

具强制条件的向量均衡问题

研究锥伪单调、锥拟凸和上锥连续映射在某种强制性条件下的向量均衡问题解集的特征,建立强制性条件与向量均衡问题解集的关系,得到对偶向量均衡问题局部解集含于向量均衡问题解集的性质和向量均衡问题解集的非空性条件,给出在锥伪单调、锥拟凸和上锥连续映射条件下向量均衡问题解集的非空有界性与强制性条件的等价性．     

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.     

向量值映射D-η-预不变真拟凸的性质

引入了向量值映射的D-η- 预不变真拟凸等概念,在下D-半连续和上D-半连续条件下分别得到了向量值映射的D-η- 预不变真拟凸的等价命题,并讨论了向量值映射的D-η- 预不变真拟凸、D-η- 严格预不变真拟凸、D-η- 半严格预不变真拟凸的关系,证明了在一定条件下,向量优化问题(VP)的局部弱有效解一定是(VP)的全局弱有效解,这些结果推广了前人所得的相应结果。     

关于一类向量极值问题弱有效解的充分必要条件

本文研究带集合约束的向量极值问题。运用局部凸Hausdorff拓扑向量空间中广义次似凸映射的择一定理和其他一些结论，得到了关于集合约束向量极值问题弱有效解的几个充分必要条件．     

非光滑多目标广义本性凸规划的最优性条件与对偶

In this paper, we introduce generalized essentially pseudoconvex function and generalized essentially quasiconvex function, and give sufficient optimality conditions of the nonsmooth generalized convex multi-objective programming and its saddle point theorem about cone efficient solution. We set up Mond-Weir type duality and Craven type duality for nonsmooth multiobjective programming with generalized essentially convex functions, and prove them.     

Stable zero duality gaps in convex programming: Complete dual characterisations with applications to semidefinite programs

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality.     

Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems

We consider convex semiinfinite programming (SIP) problems with an arbitrary fixed index set T. The article analyzes the relationship between the upper and lower semicontinuity (lsc) of the optimal value function and the optimal set mapping, and the so-called Hadamard well-posedness property (allowing for more than one optimal solution). We consider the family of all functions involved in some fixed optimization problem as one element of a space of data equipped with some topology, and arbitrary perturbations are premitted as long as the perturbed problem continues to be convex semiinfinite. Since no structure is required for T, our results apply to the ordinary convex programming case. We also provide conditions, not involving any second order optimality one, guaranteeing that the distance between optimal solutions of the discretized subproblems and the optimal set of the original problem decreases by a rate which is linear with respect to the discretization mesh-size.     

Approximate Optimality and Approximate Duality for Quasi Approximate Solutions in Robust Convex Semidefinite Programs

In this paper, we study quasi approximate solutions for a convex semidefinite programming problem in the face of data uncertainty. Using the robust optimization approach (worst-case approach), approximate optimality conditions and approximate duality theorems for quasi approximate solutions in robust convex semidefinite programming problems are explored under the robust characteristic cone constraint qualification. Moreover, some examples are given to illustrate the obtained results.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

序扰动多目标规划的锥次微分稳定性

对于局部凸拓扑向量空间的多目标规划问题,本文研究并得到当确定空间序的控制锥受扰动,它们的锥有效点（解）集和锥弱有效点（解）集分别在锥次微分和锥弱次微分意义下的稳定性结果.     

A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems

A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.     

集值映射多目标规划的K-T最优性条件

讨论集值映射多目标规划(VP)的最优性条件问题.首先,在没有锥凹的假设下,利用集值映射的相依导数,得到了(VP)的锥--超有效解要满足的必要条件和充分条件.其次,在锥凹假设和比推广了的Slater规格更弱的条件下,给出了(VP)关于锥--超有效解的K--T型最优性必要条件和充分条件.     

向量值最优化问题的最优性条件与对偶性

本文我们首先给出一类向量值优化问题（VP）的正切锥真有效解的定义，在锥方向导数的假设下，讨论了一类单目标问题 的最优性必要条件；然后利用正切锥方向导数定义一类正切锥F－凸函数类，并给出了（VP）正切锥真有效解的充分性条件，最后我们亦讨论了（VP）在正切锥真有效解意义下的对偶性质。