INTERVAL ADJUSTABLE ENTROPY ALGORITHM FOR A CLASS OF UNCONSTRAINED DISCRETE MINIMAX PROBLEMS

In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C 1.The paper deals with this problem by means of taking the place of maximum entropy function with adjustable entropy function.By constructing an interval extension of adjustable entropy function an d some region deletion test rules,a new interval algorithm is presented.The rele vant properties are proven.The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum entropy algorithm.Both theoretical and numerica l results show that the method is reliable and efficient.

Nonsmooth Equations of K-T Systems for a Constrained Minimax Problem

Using K-T optimality condition of nonsmooth optimization, we establish two equivalent systems of the nonsmooth equations for the constrained minimax problem directly. Then generalized Newton methods are applied to solve these systems of the nonsmooth equations. Thus a new approach to solving the constrained minimax problem is developed.

Solving constrained minimax problem via nonsmooth equations method

A new nonsmooth equations model of constrained minimax problem was de-rived. The generalized Newton method was applied for solving this system of nonsmooth equations system. A new algorithm for solving constrained minimax problem was established. The local superlinear and quadratic convergences of the algorithm were discussed.

Interval Algorithm for Inequality Constrained Discrete Minimax Problems

An interval algorlthm for inequality coustrained discrete minimax problems is described, in which the constrained and objective functions are C1 functions. First, based on the penalty function methods, we trans form this problem to unconstrained optimization. Second, the interval extensions of the penalty functions and the test rules of region deletion are discussed. At last, we design an interval algorithm with the bisection rule of Moore. The algorithm provides bounds on both the minimax value and the localization of the minimax points of the problem. Numerical results show that algorithm is reliable and efficiency.

A Nonmonotone Filter Method for Minimax Problems

In this paper, we propose a modified trust-region filter method algorithm for Minimax problems, which based on the framework of SQP-filter method and associated with the technique of nonmonotone method. We use the SQP subproblem to acquire an attempt step, and use the filter to weigh the effect of the attempt step so as to avoid using penalty function. The algorithm uses the Lagrange function as a merit function and the nonmonotone filter to improve the effect of the algorithm. Under some mild conditions, we prove the global convergence.

AN ITERATIVE METHOD FOR THE MINIMAX PROBLEM

In this paper a class of iterative methods for the minimax problem i; proposed.We present a sequence of the extented linear-quadratic programming (ELQP) problems as subproblems of the original minimal problem and solve the ELQP problem iteratively.The locally linear and su-perlinear convergence results of the algorithm are established.

A SSLE-TYPE ALGORITHM OF QUASI-STRONGLY SUB-FEASIBLE DIRECTIONS FOR INEQUALITY CONSTRAINED MINIMAX PROBLEMS

In this paper,we discuss the nonlinear minimax problems with inequality constraints.Based on the stationary conditions of the discussed problems,we propose a sequential systems of linear equations(SSLE)-type algorithm of quasi-strongly sub-feasible directions with an arbitrary initial iteration point.By means of the new working set,we develop a new technique for constructing the sub-matrix in the lower right corner of the coefficient matrix of the system of linear equations(SLE).At each iteration,two systems of linear equations(SLEs)with the same uniformly nonsingular coefficient matrix are solved.Under mild conditions,the proposed algorithm possesses global and strong convergence.Finally,some preliminary numerical experiments are reported.

A New Superlinearly Convergent SQP Algorithm for Nonlinear Minimax Problems

In this paper, the nonlinear minimax problems are discussed. By means of the Sequential Quadratic Programming （SQP）, a new descent algorithm for solving the problems is presented. At each iteration of the proposed algorithm, a main search direction is obtained by solving a Quadratic Programming （QP） which always has a solution. In order to avoid the Maratos effect, a correction direction is obtained by updating the main direction with a simple explicit formula. Under mild conditions without the strict complementarity, the global and superlinear convergence of the algorithm can be obtained. Finally, some numerical experiments are reported.

Error Bounds of Two Smoothing Approximations for Semi-infinite Minimax Problems

In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., the one dimensional semi-infinite minimax problems）, the primary focus of this paper is on multi- dimensional semi-infinite minimax problems. The global error bounds of two smoothing approximations for the objective function are given and compared. It is proved that the smoothing approximation given in this paper can provide a better error bound than the existing one in literature.

A New Objective Penalty Function Approach for Solving Constrained Minimax Problems

In this paper,a new objective penalty function approach is proposed for solving minimax programming problems with equality and inequality constraints.This new objective penalty function combines the objective penalty and constraint penalty.By the new objective penalty function,a constrained minimax problem is converted to minimizations of a sequence of continuously differentiable functions with a simple box constraint.One can thus apply any efficient gradient minimization methods to solve the minimizations with box constraint at each step of the sequence.Some relationships between the original constrained minimax problem and the corresponding minimization problems with box constraint are established.Based on these results,an algorithm for finding a global solution of the constrained minimax problems is proposed by integrating the particular structure of minimax problems and its global convergence is proved under some conditions.Furthermore,an algorithm is developed for finding a local solution of the constrained minimax problems,with its convergence proved under certain conditions.Preliminary results of numerical experiments with well-known test problems show that satisfactorilyapproximate solutions for some constrained minimax problems can be obtained.

AN EFFICIENT APPROACH TO NONLINEAR MINIMAX PROBLEMS

A nonlinear minimax problem is usually defined aswherefi(x), i=1,…,m, are generally smooth nonlinear functions of a vector x ∈ R^n. Since the objective φ(x) is a non-smooth function, (A) is then a non-smooth, unconstrained optimization problem and cannot be solved by standard unconstrained minimization algorithms. One normally transforms it into an equivalent nonlinear programming problem:

Entropy method for one sort of nonlinear minimax problem

In non-smooth optimization,one particular problem which often appears inengineering designs,electrical engineering and game theory is called nonlinear minimaxproblem.For the non-smooth properties of objective functions,there are some difficultiesin solving this problem.Since 1987,taking into account the entropy funtions,experts havehad several excellent results such as refs.[1—5].However,those methods are limited

MAXIMUM ENTROPY METHODS FOR CONSTRAINED OPTIMIZATION AND MINIMAX PROBLEMS

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一类无约束离散minimax问题的区间算法

讨论了一类由二阶连续可微函数构成的无约束离散 minimax问题 .通过建立极大函数的区间扩张 ,并将无解区域的凹凸性检验原则和区间 Newton迭代法引入这类特殊不可微优化问题中 ,建立了区间算法 ,证明了算法的收敛性及相关结论 ,给出了数值算例 .用该方法可以同时求出问题的 minimax值和全部 minimax点 。

解一类非线性Minimax问题

本文利用区间方法有效地解决了如下一类特殊的非线性minimax问题: F~*= F(x~*)=min max{f_1(y),f_2(y),…,f_m(y)},其中Ω_(x,η)={y|x_i-ηδ_i≤y_i≤x_i+ηδ_i,η≥0,i=1,2,…,n},公差向量δ=(δ_1,δ_2,…,δ_n)~T,δ_i>0,i=1,2,…,n。

Minimax神经网络收敛性分析

ｍｉｎｉｍａｘ问题的研究不仅在对策论、数学规划和最优控制中具有重要意义，而且许多类型的问题都需要寻求ｍｉｎｉｍａｘ问题的数值解．本文建立了连续动力系统神经网络来探讨ｍｉｎ－ｉｍａｘ问题，在适当的条件下利用Ｌｙａｐｕｎｏｖ函数讨论了网络的稳定性和收敛性，并证明了神经网络的稳定平衡点即为ｍｉｎｉｍａｘ问题的鞍点．这样的网络可由ＶＬＳＩ技术实现，且具有实时动力学行为，它们也很象生物处理的动力学．在适当的条件下，利用Ｌｙａｐｕｎｏｖ函数稳定性理论证明了该网络是Ｌｙａｐｕｎｏｖ稳定的。

求解连续Minimax问题的粒子群优化方法

Minimax问题是一类十分重要同时也是比较困难的优化问题。提出了一种基于粒子群优化的连续minimax问题求解 方法。方法的基本思想是维持两个在不同搜索空间中不对称共同进化的群体并采用粒子群优化算法获得原minimax问题的 一个解。仿真结果显示此方法可以有效求解对称和非对称连续minimax问题。

求解约束Minimax问题的神经网络模型

本文提出一种求解约束二次Minimax问题的神经网络模型 ,给出了它的Lyapunov能量函数 ,运用LaSalle不变性原理证明了它的大范围渐近稳定性 ,作为应用考察了L1范数极小化问题 .

约束Minimax问题的SQP-Filter算法及收敛性

提出了一个求解带等式和不等式约束的Minimax问题的SQP-Filter算法,每步通过求解2个二次规划子问题来得到搜索方向,并沿该方向做线搜索。该算法避免了较难的罚因子的选取,克服了Maratos效应,并在适当的假设条件下,得到了算法的全局收敛性。

单边相对光滑非凸-凹极小极大问题的镜像梯度算法

本文提出了一种镜像梯度下降梯度上升算法来求解单边相对光滑的非凸-凹极小极大问题。在算法的每次迭代中,我们采用镜像梯度下降步来更新相对光滑的变量,采用梯度上升投影步来更新目标函数中光滑的变量。本文在理论上证明了算法收敛到ε-近似一阶稳定点的迭代复杂度是O(ε^(-4))。