Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.     

Periodic solutions for second order systems at resonance

The existence and multiplicity results are obtained for periodic solutions of second order systems at resonance with unbounded nonlinearity. The proofs rely on the minimax methods and an interesting integral inequality.     

广义拟凹映射和向量极小极大不等式

引入z-广义c-拟凹向量映射,研究它和广义KKM映射的等价性,应用这些结果改进了-Ky Fan向量极小极大不等式,并且获得了一强向量极小极大不等式．     

Inequality systems and minimax theorems

This paper is a comprehensive study of the results about the consistency of inequality systems. The minimax theorems are the main tools used in proceeding to this survey. Most of the known results are quoted, and some of them are weakened or generalized to infinite dimensional spaces. Applications to programming are given.     

Standing wave solutions for a class of nonhomogeneous systems in dimension two

In this article, minimax procedures and a Trudinger–Moser type inequality in weighted Sobolev spaces are employed to establish sufficient conditions for the existence of solutions for a class of nonhomogeneous elliptic systems involving nonlinear Schrödinger equations with subcritical or critical exponential growth and with potentials which are singular and/or vanishing. The solutions are obtained by suitable control of the size of the perturbations.     

Minimax principles for elliptic mixed hemivariational–variational inequalities

In this paper, minimax principles are explored for elliptic mixed hemivariational–variational inequalities. Under certain conditions, a saddle-point formulation is shown to be equivalent to a mixed hemivariational–variational inequality. While the minimax principle is of independent interest, it is employed in this paper to provide an elementary proof of the solution existence of the mixed hemivariational–variational inequality. Theoretical results are illustrated in the applications of two contact problems.     

Some fixed-point theorems and minimax inequalities in FC-space

By applying the existence theorem of maximal elements, some new collectively fixed-point theorems for a family of set-valued mappings defined on the product space of noncompact FC-space are proved and some new theorems about minimax inequality involving two functions are given to show the relations of fixed-point theorem and minimax inequality in FC-spaces. These results improve and generalize many important results in the recent literature.     

A minimax theorem for vector-valued functions in lexicographic order

In this paper, a minimax theorem and a saddle point theorem are obtained for vector-valued functions in the sense of lexicographic order, respectively. An equivalent relationship between the minimax inequality and the saddle point is established. Some examples are given to illustrate our results.     

On a generalized sup-inf problem

In this paper, necessary and sufficient conditions for solvability of nonlinear inequality systems are given using certain generalized convexity concepts. Our results imply some theorems of Kirszbraun, Fan, Minty, Simons, Sebestyén, and Gwinner-Oettli.The authors are grateful to the referees for their helpful comments. They thank one of the referees, who emphasized the connection between the Wald minimax theorem and Theorem 2.1, and suggested an alternative proof of Theorem 2.1 in Section 4.     

一类约束不可微优化问题的极大熵方法

1.引言 用极大熵原理可以有效地处理某些优化问题,一般迭代2—6次即可达到工程要求的精度。本文给出一类约束不可微优化问题的两种极大熵方法,推广了[1,2]的结果,并研制了计算程序。试算结果说明效果良好。进一步的结果在[4]中给出。 考虑下述问题:     

精确罚函数和极小极大问题

在这篇文章中我们研究了对于不等式约束的非线性规划问题如何根据极小极大问题的鞍点来找精确罚问题的解。对于一个具有不等式约束的非线性规划问题,通过罚函数,我们构造出一个极小极大问题,应用交换“极小”或“极大”次序的策略,证明了罚问题的鞍点定理。研究结果显示极小极大问题的鞍点是精确罚问题的解。     

General KKM Theorem with Applications to Minimax and Variational Inequalities

In this paper, a general version of the KKM theorem is derived by using the concept of generalized KKM mappings introduced by Chang and Zhang. By employing our general KKM theorem, we obtain a general minimax inequality which contains several existing ones as special cases. As applications of our general minimax inequality, we derive an existence result for saddle-point problems under general setting. We also establish several existence results for generalized variation inequalities and generalized quasi-variational inequalities.     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

不等式约束极大极小问题的一个新型模松弛强次可行SQCQP算法

针对带不等式约束的极大极小问题,借鉴一般约束优化问题的模松弛强次可行SQP算法思想,提出了求解不等式约束极大极小问题的一个新型模松弛强次可行SQCQP算法.首先,通过在QCQP子问题中选取合适的罚函数,保证了算法的可行性以及目标函数F(x)的下降性,同时简化QCQP子问题二次约束项参数α_k的选取,可保证算法的可行性和收敛性.其次,算法步长的选取合理简单.最后,在适当的假设条件下证明了算法具有全局收敛性及强收敛性.初步的数值试验结果表明算法是可行有效的.     

Minimax inequalities equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz theorems

The purpose of this note is to give further generalizations of the Ky Fan minimax inequality by relaxing the compactness and convexity of sets and the quasi-concavity of the functional and to show that our minimax inequalities are equivalent to the Fan-Knaster-Kuratowski-Mazurkiewicz (FKKM) theorem and a modified FKKM theorem given in this note.     

非紧L-凸空间中的一个新的极大元定理及其对鞍点的应用(英文)

在非紧L-凸空间中建立一个新的极大元定理.作为应用,获得了非紧L-凸空间中的极大极小不等式和鞍点定理.     

Two minimax inequalities in G-convex spaces

In this work we obtain two minimax inequalities in $G$-convex spaces which extend and improve a large number of generalizations of the Ky Fan minimax inequality and of the von Neumann–Sion minimax principle.     

A solvability theorem and minimax fractional programming

This paper presents a stable solvability theorem for general inequality systems under a local closedness condition. It is shown how this mild regularity condition can be characterized by the validity of the solvability theorem for all local perturbations. Based on this solvability theorem zero duality gap and stability are established for general minimax fractional programming problems.The research was initiated while the first named author was a visitor at the University of New South Wales and was completed while the second named author was a visitor at the Technische Hochschule Darmstadt.     

Indirect Obstacle Minimax Control for Elliptic Variational Inequalities

A minimax control problem for a coupled system of a semilinear elliptic equation and an obstacle variational inequality is considered. The major novelty of such problem lies in the simultaneous presence of a nonsmooth state equation (variational inequality) and a nonsmooth cost function (sup norm). In this paper, the existence of optimal controls and the optimality conditions are established.     

Minimax Theorems with Bregman Distance in Banach Spaces

Since Bregman distance is a generalized distance function which does not satisfy the triangle inequality nor symmetry, it is important to study minimax theorems in reflexive Banach spaces. Our minimax theorems differ from those in previous literature.