On a class of functions with equal infima over a domain and its boundary

In this paper, we present a class of functions:f:X such that inf _xX f(x)= , whereX is a nonempty, finitely compact and convex set in a vector space andB _x ={xX: y aff(X){x:[x, y]X={x}. Our main tool is a recent minimax theorem by Ricceri (Ref. 1).     

A smooth method for the finite minimax problem

We consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions. We prove that, under usual assumptions, it is possible to construct a continuously differentiable function, whose minimizers yield the minimizers of the max function and the corresponding minimum values. On this basis, we can define implementable algorithms for the solution of the minimax problem, which are globally convergent at a superlinear convergence rate. Preliminary numerical results are reported.This research was partially supported by the National Research Program on Metodi di ottimizzazione per le decisioni, Ministero dell'Università e della Ricerca Scientifica e Tecnologica, Italy.     

Convexlike and concavelike conditions in alternative,minimax, and minimization theorems

Convexlike and concavelike conditions are of interest for extensions of the Von Neumann minimax theorem. Since the beginning of the 80's, these conditions also play a certain role in deriving generalized alternative theorems of the Gordan, Motzkin, and Farkas type and Lagrange multiplier results for constrained minimization problems.In this paper, we study various known convexlike conditions for vector-valued functions on a set and investigate convexlike and concavelike conditions for real-valued functions on a product setC×D, where we are mainly interested in the relationships between these conditions. At the end of the paper, we point out several conclusions from our results for the above-mentioned mathematical fields.The author is indebted to Dr. R. Reemtsen and Dr. V. Jeyakumar for their helpful comments during the preparations of this paper.     

Numerical solution of minimax problems of optimal control,part 1

This paper contains general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We consider two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).For Problem (Q), we exploit the analogy with a bounded-state problem in combination with a transformation of the Jacobson type. This requires the proper augmentation of the state vectorx(t), the control vectoru(t), and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.For Problem (R), we exploit the analogy with a bounded-control problem in combination with a transformation of the Valentine type. This requires the proper augmentation of the control vectoru(t) and the parameter vector , as well as the proper augmentation of the constraining relations. As a result of the transformation, the unknown minimax value of the performance index becomes a component of the parameter vector being optimized.In a subsequent paper (Part 2), the transformation techniques presented here are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer; both the single-subarc approach and the multiple-subarc approach are discussed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

拟线性椭圆方程共振问题解的存在定理

考虑具有无界非线性项的椭圆方程在任意特征值的共振问题. 运用临界点理论中的极小极大方法得到了边值问题-Δpu =λ| u |p-2u g(u) - f(x)　　　在Ω内u =0　　　　　　　　　　　　　在Ω上的解.     

求解极大极小问题的调节熵函数法的收敛性

对调节熵函数法的收敛性作了理论分析.     

解非线性极大极小问题的格雷码加速遗传算法

引进求解非线性极大极小问题的格雷码加速遗传算法（GAGA），给出GAGA算法实施的详细步骤，建立了GAGA相应的收敛定理。对GAGA的有效性和可行性进行了理论分析和实例分析。与一般的格雷码遗传算法（GGA）相比，GAGA具有准确、快速和适用性强等特点，是一种既可以较大概率搜索全局最优解，又能进行局部细致搜索的优秀非线性优化方法，可广泛应用于各种优化问题中。     

非光滑极小极大神经网络收敛性分析

利用非光滑分析的理论讨论了非可微鞍泛函的ｍｉｎｉｍａｘ问题，并建立非光滑神经网络来求鞍泛函的鞍点，在适当的条件下，利用Ｌｙａｐｕｎｏｖ理论讨论了网络的收敛性怀稳定性。     

弱Fan Ky点与不具连续性条件的函数的极大极小不等式定理

基于弱Fan Ky点的概念，不具任何连续性条件的函数弱Fan Ky点的存在性，给出了几个极大极小不等式定理，所得结果将著名的Fan Ky不等式定理推广至不具连续性条件的情形。     

Properties of ε-value functions for a two person differential game

We consider the problem of approximate minimax for the Bolza problem of optimal control. Starting from the method of dynamic programming (Bellman) we define the ?-value function to be the approximation for the value function being a solution to the Hamilton–Jacobi equation.