一类反凸规划的全局新算法

§1.引言 到目前为止,大多数非线性规划的有效算法都是寻求它的局部最优解,由于很难判断一个局部解是否就是一个全局解,全局规划的研究是个困难问题,反凸规划由于其可行域的非凸性甚至非连通性,目前有效算法更少。 [1]已经指出很容易把D.C.规划(即目标函数和约束函数均为二个凸函数之差)转化成为一个目标函数为线性的反凸规划:     

一类线性乘积规划问题的分支定界缩减方法

提出了求解一类线性乘积规划问题的分支定界缩减方法, 并证明了算法的收敛性.在这个方法中, 利用两个变量乘积的凸包络技术, 给出了目标函数与约束函数中乘积的下界, 由此确定原问题的一个松弛凸规划, 从而找到原问题全局最优值的下界和可行解. 为了加快所提算法的收敛速度, 使用了超矩形的缩减策略. 数值结果表明所提出的算法是可行的.     

一些类型的数学规划问题的全局最优解

本文对严格单调函数给出了几个凸化和凹化的方法，利用这些方法可将一个严格单调的规划问题转化为一个等价的标准D．C．规划或凹极小问题．本文还对只有一个严格单调的约束的非单调规划问题给出了目标函数的一个凸化和凹化方法，利用这些方法可将只有一个严格单调约束的非单调规划问题转化为一个等价的凹极小问题．再利用已有的关于D．C．规划和凹极小的算法，可以求得原问题的全局最优解．     

凸多乘积问题的完全多项式时间近似算法

针对凸多乘积问题,提出一种求其全局最优解的近似算法.首先,通过引入参量获得一个等价问题,然后估计问题中每一乘积项的上下界,进而借助网格结点,获得一些凸规划问题,通过求解这些凸规划问题获得原问题的近似最优解.最后,给出了该算法的收敛性证明和计算复杂性分析.     

广义几何规划的全局优化算法

对许多工程设计中常用的广义几何规划问题(GGP)提出一种确定性全局优化算法,该算法利用目标和约束函数的线性下界估计,建立GGP的松弛线性规划(RLP),从而将原来非凸问题(GGP)的求解过程转化为求解一系列线性规划问题(RLP).通过可行域的连续细分以及一系列线性规划的解,提出的分枝定界算法收敛到GGP的全局最优解,且数值例子表明了算法的可行性.     

一种整数线性乘积规划问题的分支定界算法

本文为了求解整数线性乘积规划(ILMP)问题的全局最优解,提出一种新的线性松弛分支定界算法.该算法利用对数函数的单调性及凹凸性,得到(ILMP)全局最小值的下界,并利用区域缩减技术以最大限度地删除不可行区域,加快该算法的收敛速度.最后数值实验表明,本文提出的算法是有效并且可行的.     

线性比式和规划问题的输出空间分支定界算法

本文旨在针对线性比式和规划这一NP-Hard非线性规划问题提出新的全局优化算法.首先,通过引入p个辅助变量把原问题等价的转化为一个非线性规划问题,这个非线性规划问题的目标函数是乘积和的形式并给原问题增加了p个新的非线性约束,再通过构造凸凹包络的技巧对等价问题的目标函数和约束条件进行相应的线性放缩,构成等价问题的一个下界线性松弛规划问题,从而提出了一个求解原问题的分支定界算法,并证明了算法的收敛性.最后,通过数值结果比较表明所提出的算法是可行有效的.     

一类多乘积规划的加速算法

针对一类多乘积规划问题(MP),给出一个加速算法.首先导出一个与(MP)等价的逆凸问题(RCP),然后构造问题(RCP)的线性松弛化问题.算法的主要特点是提出了两个加速技巧,这些技巧可以用于改善算法的收敛速度.数值算例表明算法是可行的.     

全局求解线性多乘积规划的分支定界算法

本文针对一类线性多乘积规划问题提出一种分支定界算法.首先将原问题转化为其等价形式,然后利用提出的线性松弛技术将等价问题松弛为线性规划问题,通过求解一系列线性规划问题得到原问题的全局最优解.最后给出算法的收敛性和计算复杂性.数值实验表明算法是有效的.     

求线性比试和问题的确定性全局优化算法

为求线性比试和问题的全局最优解,本文给出了一个分支定界算法.通过一个等价问题和一个新的线性化松弛技巧,初始的非凸规划问题归结为一系列线性规划问题的求解.借助于这一系列线性规划问题的解,算法可收敛于初始非凸规划问题的最优解.算法的计算量主要是一些线性规划问题的求解.数值算例表明算法是切实可行的.     

A branch and bound algorithm for globally solving a class of nonconvex programming problems

A branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems (NP). For minimizing the problem, linear lower bounding functions (LLBFs) of objective function and constraint functions are constructed, then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value. The proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems. And finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm.     

Parametric simplex algorithms for solving a special class of nonconvex minimization problems

It is shown that parametric linear programming algorithms work efficiently for a class of nonconvex quadratic programming problems called generalized linear multiplicative programming problems, whose objective function is the sum of a linear function and a product of two linear functions. Also, it is shown that the global minimum of the sum of the two linear fractional functions over a polytope can be obtained by a similar algorithm. Our numerical experiments reveal that these problems can be solved in much the same computational time as that of solving associated linear programs. Furthermore, we will show that the same approach can be extended to a more general class of nonconvex quadratic programming problems.     

边界约束非凸二次规划问题的分枝定界方法

本文是研究带有边界约束非凸二次规划问题,我们把球约束二次规划问题和线性约束凸二次规划问题作为子问题,分明引用了它们的一个求整体最优解的有效算法,我们提出几种定界的紧、松驰策略,给出了求解原问题整体最优解的分枝定界算法,并证明了该算法的收敛性,不同的定界组合就可以产生不同的分枝定界算法,最后我们简单讨论了一般有界凸域上非凸二次规划问题求整体最优解的分枝与定界思想。     

A global optimization using linear relaxation for generalized geometric programming

Many local optimal solution methods have been developed for solving generalized geometric programming (GGP). But up to now, less work has been devoted to solving global optimization of (GGP) problem due to the inherent difficulty. This paper considers the global minimum of (GGP) problems. By utilizing an exponential variable transformation and the inherent property of the exponential function and some other techniques the initial nonlinear and nonconvex (GGP) problem is reduced to a sequence of linear programming problems. The proposed algorithm is proven that it is convergent to the global minimum through the solutions of a series of linear programming problems. Test results indicate that the proposed algorithm is extremely robust and can be used successfully to solve the global minimum of (GGP) on a microcomputer.     

A Branch and Bound Algorithm for Solving Low Rank Linear Multiplicative and Fractional Programming Problems

This paper is concerned with a practical algorithm for solving low rank linear multiplicative programming problems and low rank linear fractional programming problems. The former is the minimization of the sum of the product of two linear functions while the latter is the minimization of the sum of linear fractional functions over a polytope. Both of these problems are nonconvex minimization problems with a lot of practical applications. We will show that these problems can be solved in an efficient manner by adapting a branch and bound algorithm proposed by Androulakis–Maranas–Floudas for nonconvex problems containing products of two variables. Computational experiments show that this algorithm performs much better than other reported algorithms for these class of problems.     

Column enumeration based decomposition techniques for a class of non-convex MINLP problems

We propose a decomposition algorithm for a special class of nonconvex mixed integer nonlinear programming problems which have an assignment constraint. If the assignment decisions are decoupled from the remaining constraints of the optimization problem, we propose to use a column enumeration approach. The master problem is a partitioning problem whose objective function coefficients are computed via subproblems. These problems can be linear, mixed integer linear, (non-)convex nonlinear, or mixed integer nonlinear. However, the important property of the subproblems is that we can compute their exact global optimum quickly. The proposed technique will be illustrated solving a cutting problem with optimum nonlinear programming subproblems.     

Global Optimization of Nonconvex Polynomial Programming Problems Having Rational Exponents

This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.     

Image space analysis of generalized fractional programs

The solution of a particular nonconvex program is usually very dependent on the structure of the problem. In this paper we identify classes of nonconvex problems involving either sums or products of ratios of linear terms which may be treated by analysis in a transformed space. In each class, the image space is defined by a mapping which associates a new variable with each original ratio of linear terms. In the image space, optimization is easy in certain directions, and the overall solution may be realized by sequentially optimizing in these directions.In addition to these ratio problems, we also show how to use image space analysis to treat the subclass of problems whose objective is to optimize a product of linear terms. For each class of nonconvex problems, we present an algorithm that locates global solutions by computing both upper and lower bounds on the solution and then solving a sequence of linear programming sub-problems. We also demonstrate the algorithms described in this paper by solving several example problems.     

Global optimization of nonconvex factorable programming problems

In this paper, we consider a special class of nonconvex programming problems for which the objective function and constraints are defined in terms of general nonconvex factorable functions. We propose a branch-and-bound approach based on linear programming relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev interpolation polynomials coordinated with a Reformulation-Linearization Technique (RLT). A suitable partitioning process is proposed that induces convergence to a global optimum. The algorithm has been implemented in C++ and some preliminary computational results are reported on a set of fifteen engineering process control and design test problems from various sources in the literature. The results indicate that the proposed procedure generates tight relaxations, even via the initial node linear program itself. Furthermore, for nine of these fifteen problems, the application of a local search method that is initialized at the LP relaxation solution produced the actual global optimum at the initial node of the enumeration tree. Moreover, for two test cases, the global optimum found improves upon the solutions previously reported in the source literature. Received: January 14, 1998 / Accepted: June 7, 1999?Published online December 15, 2000