多目标半定规划的互补弱鞍点和G-鞍点最优性条件

对于含矩阵函数半定约束和多个目标函数的多目标半定规划问题，给出Lagrange函数在弱有效意义下的互补弱鞍点和Geofrrion恰当有效意义下的G-鞍点的定义及其等价定义．然后，在较弱的凸性条件下，利用含矩阵和向量约束的择一性定理，建立多目标半定规划的互补弱鞍点和G-鞍点充分必要条件．     

集值映射多目标半定规划问题的epsilon-弱有效性

对于集值映射多目标半定规划问题, 在近似锥-次类凸的框架下, 建立了含矩阵和向量的择一性定理, 给出了问题的epsilon-弱有效解的epsilon-Lagrange乘子定理及标量化定理和epsilon-弱鞍点定理．     

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。     

集值映射向量优化问题的ε-超鞍点和ε-对偶定理

本文研究集值映射向量优化问题的ε-超鞍点和ε-对偶定理。在集值映射是近似广义锥次似凸的假设下，利用ε-超有效解的标量化和Lagrange乘子定理，建立和证明了关于ε-超有效解的鞍点和对偶定理。     

集值映射向量优化问题的e-真有效解

引入了集值映射向量优化问题的αe-弱有效解、e-真有效解、e-真鞍点概念,在近似广义C-次似凸条件下,建立了e-真有效解的标量化定理、Lagrang乘子定理和e-真鞍点定理,并讨论了集值映射向量优化问题的αe-弱有效解的标量化定理和Laugrange乘子定理,推广了已有结果。     

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.     

集值映射多目标半定规划问题的ε-弱有效性

对于集值映射多目标半定规划问题,在近似锥.次类凸的框架下,建立了含矩阵和向量的择一性定理,给出了问题的ε-弱有效解的ε-Lagrange乘子定理及标量化定理和ε-弱鞍点定理.     

群体多目标最优化的Lagrange对偶性

本文建立了目标和约束为不对称的群体多目标最优化问题的Lagrange对偶规划，在问题的联合弱有效解意义下，得到群体多目标最优化Lagrange型的弱对偶定理、基本对偶定理、直接对偶定理和逆对偶定理。     

多目标优化ε-拟弱有效解的最优性条件

本文研究了多目标优化问题的ε-拟弱有效解.通过Hadamard(上、下)方向导数和极限次微分给出了ε-拟弱有效解存在的充分和必要条件.提出了一种ε-拟弱鞍点的概念,给出了鞍点存在的条件.最后,建立了拉格朗日对偶模型,证明了对偶定理.     

非光滑半无限多目标规划近似解的对偶和鞍点定理

本文利用切向次微分研究了一类非光滑半无限多目标规划问题,并讨论了它的对偶定理和鞍点定理.首先,建立了半无限多目标规划问题的Mond-Weir型对偶,在广义凸性假设下,获得了半无限多目标规划问题近似解的弱对偶、强对偶和逆对偶定理.其次,定义了向量值拉格朗日函数的ε-拟鞍点,获得了ε-拟鞍点的必要和充分条件.这些结论推广和改进了文献中的相应结果.最后以具体的例子来说明了本文的结论.     

ε-Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints

Using a scalarization method, approximate optimality conditions of a multiobjective nonconvex optimization problem which has an infinite number of constraints are established. Approximate duality theorems for mixed duality are given. Results on approximate duality in Wolfe type and Mond-Weir type are also derived. Approximate saddle point theorems of an approximate vector Lagrangian function are investigated.     

On optimality and duality theorems of nonlinear disjunctive fractional minmax programs

This paper is concerned with the study of optimality conditions for disjunctive fractional minmax programming problems in which the decision set can be considered as a union of a family of convex sets. Dinkelbach’s global optimization approach for finding the global maximum of the fractional programming problem is discussed. Using the Lagrangian function definition for this type of problem, the Kuhn–Tucker saddle point and stationary-point problems are established. In addition, via the concepts of Mond–Weir type duality and Schaible type duality, a general dual problem is formulated and some weak, strong and converse duality theorems are proven.     

Φ-Conjugation and noneonvex optimization. a survey (part III)

In the present paper, which is part III of our review concerning the theory of Φ-conjugate functions, we consider Lagrangians, duality theorems are proved and the connection to saddle point theorems is shown. By a fundamental inequality, duality theorems are proved and the connection to saddle point theorems is shown. By a fundamental inequality, duality theorems can be obtained, where results are modified given in part I and part II of our paper.     

On the optimality of some classes of invex problems

In this paper we define two notions: Kuhn–Tucker saddle point invex problem with inequality constraints and Mond–Weir weak duality invex one. We prove that a problem is Kuhn–Tucker saddle point invex if and only if every point, which satisfies Kuhn–Tucker optimality conditions forms together with the respective Lagrange multiplier a saddle point of the Lagrange function. We prove that a problem is Mond–Weir weak duality invex if and only if weak duality holds between the problem and its Mond–Weir dual one. Additionally, we obtain necessary and sufficient conditions, which ensure that strong duality holds between the problem with inequality constraints and its Wolfe dual. Connections with previously defined invexity notions are discussed.     

Vector continuous-time programming without differentiability

In this work continuous-time programming problems of vector optimization are considered. Firstly, a nonconvex generalized Gordan’s transposition theorem is obtained. Then, the relationship with the associated weighting scalar problem is studied and saddle point optimality results are established. A scalar dual problem is introduced and duality theorems are given. No differentiability assumption is imposed.     

A duality approach to minimax results for quasi-saddle functions in finite dimensions

In this paper we show how saddle point theorems for a quasiconvex—quasiconcave function can be derived from duality theory. A symmetric duality framework that provides the machinery for deriving saddle point theorems is presented. Generating the theorems,via the framework, provides a deeper understanding of assumptions employed in existing theorems which do not utilize duality theory.     

Duality theories in nonlinear semidefinite programming

We consider the duality theories in nonlinear semidefinite programming. Some duality theorems are established to show the important relations among the optimal solutions and optimal values of the primal, the dual and the saddle point problems of nonlinear semidefinite programming.     

Interior penalty functions and duality in linear programming

Logarithmic additive terms of barrier type with a penalty parameter are included in the Lagrange function of a linear programming problem. As a result, the problem of searching for saddle points of the modified Lagrangian becomes unconstrained (the saddle point is sought with respect to the whole space of primal and dual variables). Theorems on the asymptotic convergence to the desired solution and analogs of the duality theorems for the arising optimization minimax and maximin problems are formulated.     

凸规划的一个对偶问题

1引言考虑标准的非可微凸规划问题     

半无限规划的对偶性

本文对半无限凸规划提出一个新的对偶问题,它由扰动函数及其次微分刻划.同时讨论了弱对偶性、强对偶性及逆对偶性,证明强对偶性等价于鞍点准则.