Necessary and sufficient conditions in constrained optimization

Additional conditions are attached to the Kuhn-Tucker conditions giving a set of conditions which are both necessary and sufficient for optimality in constrained optimization, under appropriate constraint qualifications. Necessary and sufficient conditions are also given for optimality of the dual problem. Duality and converse duality are treated accordingly.     

Kuhn-Tucker条件与线性规划的对偶性

证明了线性规划的K uhn-Tucker条件蕴含着它的对偶问题,解释了L agrange乘子的意义.进而显示了K-T条件中的互补松驰性与对偶线性规划的互补松紧定理之间的联系.     

Benson真有效意义下集值优化的广义最优性条件

本文引入了关于集值映射的α-阶Clarke切导数、α-阶邻接切导数及α-阶 伴随切导数的概念,借此建立了约束向量集值优化Benson真有效解导数型的Kuhn- Tucker条件.     

Sufficient optimality conditions and duality in vector optimization with invex-convexlike functions

We prove the Kuhn-Tucker sufficient optimality condition, the Wolfe duality, and a modified Mond-Weir duality for vector optimization problems involving various types of invex-convexlike functions. The class of such functins contains many known generalized convex functions. As applications, we demonstrate that, under invex-convexlikeness assumptions, the Pontryagin maximum principle is a sufficient optimality condition for cooperative differential games. The Wolfe duality is established for these games.The author is indebted to the referees and Professor W. Stadler for valuable remarks and comments, which have been used to revise considerably the paper.     

Invex-convexlike functions and duality

We define a class of invex-convexlike functions, which contains all convex, pseudoconvex, invex, and convexlike functions, and prove that the Kuhn-Tucker sufficient optimality condition and the Wolfe duality hold for problems involving such functions. Applications in control theory are given.The author is grateful to Professor W. Stadler and the referees for many valuable remarks and suggestions, which have enabled him to improve considerably the paper.     

On invexity-type nonlinear programming problems

In this paper, we propose a new class of nonlinear programing, called SFJ-invex programming. The optimality characterization shows that a problem is SFJ-invex if and only if a Fritz John point together with its multiplier, is a Fritz John saddle point of the problem. Under any constraint qualification assumption, a problem is SFJ-invex if and only if a Kuhn-Tucker point together with its multiplier is a Kuhn-Tucker saddle point of the problem. Furthermore, a generalization of the SFJ-invex, class is developed; the applications to (h, )-convex programming, particularly geometric programming, and to generalized fractional programming provide a relaxation in constraint qualification for differentiable problems to get saddle-point type optimality criteria.The author wishes to thank the referee for helpful comments.     

Nondifferentiable mathematical programming and convex-concave functions

For a convex-concave functionL(x, y), we define the functionf(x) which is obtained by maximizingL with respect toy over a specified set. The minimization problem with objective functionf is considered. We derive necessary conditions of optimality for this problem. Based upon these necessary conditions, we define its dual problem. Furthermore, a duality theorem and a converse duality theorem are obtained. It is made clear that these results are extensions of those derived in studies on a class of nondifferentiable mathematical programming problems.This work was supported by the Japan Society for the Promotion of Sciences.     

稀缺资源的节省利用研究

本在建立模型的基础上，对稀缺资源的节省利用问题进行了研究，给出了稀缺资源能否节省的判别条件和具体作法。     

Necessary and Sufficient Conditions for Weak Quasi Efficiency in Nonsmooth Multiobjective Problems

In this article, a multiobjective problem with a feasible set defined by inequality, equality and set constraints is considered, where the objective and constraint functions are locally Lipschitz. Several constraint qualifications are given and the relations between them are analyzed. We establish Kuhn-Tucker and strong Kuhn-Tucker necessary optimality conditions for (weak) quasi e?ciency in terms of the Clarke subdifferential. By using two new classes of generalized convex functions, su?cient conditions for local (weak) quasi e?cient are also provided. Furthermore, we study the Mond-Weir type dual problem and establish weak, strong and converse duality results.     

Scalarization and Kuhn-Tucker-Like Conditions for Weak Vector Generalized Quasivariational Inequalities

We consider a weak vector generalized quasivariational inequality. By introducing a method of scalarization which does not require any assumption on the data and by using previous results of the authors concerning scalar generalized quasivariational inequalities, we present Kuhn-Tucker-like conditions for this problem in the case in which the set-valued operator of the constraints is defined by a finite number of inequalities