一类特殊多项式整数规划问题的最优化算法

本文考虑了一类特殊的多项式整数规划问题。此类问题有很广泛的实际应用,并且是NP难问题。对于这类问题,最优性必要条件和最优性充分条件已经给出。我们在本文中将要利用这些最优性条件设计最优化算法。首 先,利用最优性必要条件,我们给出了一种新的局部优化算法。进而我们结合最优性充分条件、新的局部优化算法和辅助函数,设计了新的全局最优化算法。本文给出的算例展示出我们的算法是有效的和可靠的。     

On the Construction of Convex and Concave Envelope Formulas for Bilinear and Fractional Functions on Quadrilaterals

Convex and concave envelopes play important roles in various types of optimization problems. In this article, we present a result that gives general guidelines for constructing convex and concave envelopes of functions of two variables on bounded quadrilaterals. We show how one can use this result to construct convex and concave envelopes of bilinear and fractional functions on rectangles, parallelograms and trapezoids. Applications of these results to global optimization are indicated.     

A Combined Algorithm for Fractional Programming

In this paper, we present an outer approximation algorithm for solving the following problem: max _xS {f(x)/g(x)}, where f(x)0 and g(x)>0 are d.c. (difference of convex) functions over a convex compact subset S of R ⁿ . Let ()=max _xS (f(x)–g(x)), then the problem is equivalent to finding out a solution of the equation ()=0. Though the monotonicity of () is well known, it is very time-consuming to solve the previous equation, because that maximizing (f(x)–g(x)) is very hard due to that maximizing a convex function over a convex set is NP-hard. To avoid such tactics, we give a transformation under which both the objective and the feasible region turn to be d.c. After discussing some properties, we propose a global optimization approach to find an optimal solution for the encountered problem.     

Conical Partition Algorithm for Maximizing the Sum of dc Ratios

The following problem is considered in this paper: 0, j = 1,\ldots,m,$$" align="middle" border="0"> are d.c. (difference of convex) functions over a convex compact set D in R^n. Specifically, it is reformulated into the problem of maximizing a linear objective function over a feasible region defined by multiple reverse convex functions. Several favorable properties are developed and a branch-and-bound algorithm based on the conical partition and the outer approximation scheme is presented. Preliminary results of numerical experiments are reported on the efficiency of the proposed algorithm.AMS Subject Classifications: 90C32, 90C30, 65K05.The authors were partially supported by a Grant-in-Aid (Yang Dai: C-13650444; Jianming Shi and Shouyang Wang: C-14550405) of the Ministry of Education, Science, Sports and Culture of Japan.     

应用模糊目标规划方法求解多目标线性分式规划问题

针对多目标分式线性规划问题,提出利用上(下)界表示目标期望水平及允许上(下)限,且利用一阶泰勒公式逼近隶属函数,将多目标分式规划转化为线性规划问题,并用单纯形法求解,通过实验算例说明了所提出的方法的有效性.     

一类插值多项式算子与无界函数逼近

将经典“试探函数组”1,x,x^2应用于扩展乘数法,建立了一个判别线性正算子能否改造为逼近任何无界连续函数的充要条件。利用该条件给出了一类变形的插值多项式算子的收敛性定理,得到了具有一般性的结论。     

On a Polynomial Fractional Formulation for Independence Number of a Graph

In this paper we characterize the local maxima of a continuous global optimization formulation for finding the independence number of a graph. Classical Karush-Kuhn-Tucker conditions and simple combinatorial arguments are found sufficient to deduce several interesting properties of the local and global maxima. These properties can be utilized in developing new approaches to the maximum independent set problem.     

A Unified Monotonic Approach to Generalized Linear Fractional Programming

We present an efficient unified method for solving a wide class of generalized linear fractional programming problems. This class includes such problems as: optimizing (minimizing or maximizing) a pointwise maximum or pointwise minimum of a finite number of ratios of linear functions, optimizing a sum or product of such ratios, etc. – over a polytope. Our approach is based on the recently developed theory of monotonic optimization.     

线性分式规划问题的多项式时间近似算法

本文针对线性比式和分式规划问题,提出一种求其全局最优解的完全多项式时间近似算法,并从理论上证明该算法的收敛性和计算复杂性,数值算例也说明了算法是可行的.     

Complete Solutions and Extremality Criteria to Polynomial Optimization Problems

This paper presents a set of complete solutions to a class of polynomial optimization problems. By using the so-called sequential canonical dual transformation developed in the author’s recent book [Gao, D.Y. (2000), Duality Principles in Nonconvex Systems: Theory, Method and Applications, Kluwer Academic Publishers, Dordrecht/Boston/London, xviii + 454 pp], the nonconvex polynomials in can be converted into an one-dimensional canonical dual optimization problem, which can be solved completely. Therefore, a set of complete solutions to the original problem is obtained. Both global minimizer and local extrema of certain special polynomials can be indentified by Gao-Strang’s gap function and triality theory. For general nonconvex polynomial minimization problems, a sufficient condition is proposed to identify global minimizer. Applications are illustrated by several examples.     

一种新的快速逼近真值的程序算法──分布误差法

本文介绍一种新的逼近真值的快速程序算法及其应用。     

Mathematical Properties of Optimization Problems Defined by Positively Homogeneous Functions

We consider the nonlinear programming problem

Global Optimization Algorithm for the Nonlinear Sum of Ratios Problem

This article presents a branch-and-bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm economizes the required computations by conducting the branch-and-bound search in ^p, rather than in ⁿ, where p is the number of ratios in the objective function of problem (P) and n is the number of decision variables in problem (P). To implement the algorithm, the main computations involve solving a sequence of convex programming problems for which standard algorithms are available.     

Generating Sum-of-Ratios Test Problems in Global Optimization

A method is presented for the construction of test problems involving the minimization over convex sets of sums of ratios of affine functions. Given a nonempty, compact convex set, the method determines a function that is the sum of linear fractional functions and attains a global minimum over the set at a point that can be found by convex programming and univariate search. Generally, the function will have also local minima over the set that are not global minima.     

一类新的分式规划问题的全局优化方法

本文对一类新的分式规划问题(FP)提出了一个有效的全局优化方法.首先将问题(FP)转化为其等价问题(EFP),然后利用线性化技术建立了(EFP)的松弛线性规划问题(RLP),通过对其可行域的细分和求解一系列的线性规划,得到问题(EFP)的全局最优值的上下界.理论证明和数值试验的结果都表明该算法能有效求解问题(FP),推广了线性比式和的情形.     

关于无界连续函数逼近

§ 1 .　Introduction　　Itisaveryimportantstudytoapproximatenonboundedcontinuousfunctionsdefinedonalargerange.Aneffectivemethodcalledthemethodofmultiplierenlargementhasbeenputfor wardandgreatlydevelopedbyHSULCandWANGReng Hong [1— 3].Thesecondauthorofthispaperinparticularhasprovedin [2 ]almostuniformlythatanynonboundedcontinuousfunctionsdefinedonawholerealaxiscanbeapproximatedbypolynomialoperators.We’llgen eralizeinthispaperthebasicprincipleofthismethod,andasanexample,Bernsteinpolynomial…     

求全局最优化问题的一类单参数全局凸填充函数

讨论求全局最优化问题的填充函数法,进一步提出了求全局最优化问题的一类单参数全局凸填充函数,它和目标函数同阶可微.     

求广义线性比试和问题全局解的新方法

为确定广义线性比式和规划问题(GFP)的全局最优解,提出一个新的分支定界方法.在算法中,分支过程采用单纯形对分规则,且界的估计通过一些线性规划问题的求解完成.给出算法的收敛性证明.数值试验结果显示算法是有效可行的.     

一类特殊的分式规划问题的ε-近似算法

本文针对一类特殊的分式规划问题基于网格搜索提出了一个求其全局最优解的算法,且从理论上证明了算法的收敛性与计算复杂性,通过算例验证了算法的可行性与有效性.     

一类线性约束下不可微最优化问题的可行下降方法

本文给出了一类线性约束下不可微量优化问题的可行下降方法，这类问题的目标函数是凸函数和可微函数的合成函数，算法通过解系列二次规划寻找可行下降方向，新的迭代点由不精确线搜索产生，在较弱的条件下，我们证明了算法的全局收敛性