求二次比式和问题全局解的一个新的确定性算法

本文研究一类二次比式和规划问题.首先,利用等价转换的方法把原问题转化为一个非线性规划问题,并且这个非线性规划问题的目标函数通项的分子和分母都分别是两项线性函数乘积和再加上一个线性函数的形式,再根据两项线性函数乘积和的特性,对目标函数进行线性松弛,以确定原问题最优值的下界,从而提出一个求解线性规划问题的分支定界算法,并证明该算法的收敛性.最后,数值结果表明所提出的算法是可行有效的.     

基于D.C.分解的一类箱型约束的非凸二次规划的新型分支定界算法

提出了一类求解带有箱约束的非凸二次规划的新型分支定界算法.首先,把原问题目标函数进行D.C.分解(分解为两个凸函数之差),利用次梯度方法,求出其线性下界逼近函数的一个最优值,也即原问题的一个下界.然后,利用全局椭球算法获得原问题的一个上界,并根据分支定界方法把原问题的求解转化为一系列子问题的求解.最后,理论上证明了算法的收敛性,数值算例表明算法是有效可行的.     

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.     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

求一类非凸规划问题全局解的确定性算法

本文对一类非凸规划问题(NP)给出一确定性全局优化算法.这类问题包括:在非凸的可行域上极小化有限个带指数的线性函数乘积的和与差,广义线性多乘积规划,多项式规划等.通过利用等价问题和线性化技巧提出的算法收敛到问题(NP)的全局极小.     

一类DC规划问题的分支定界算法

本文针对一类带有箱子和线性不等式约束的特殊DC规划问题,提出了一种分支定界算法.首先将原问题转化为其等价问题,然后利用目标函数的特点将等价问题松弛为凸规划问题,通过求解一系列凸规划问题得到原问题的最优解,最后给出算法的收敛性证明.数值实验表明该算法是可行有效的.     

一类全局优化问题的线性化方法

本文对带有不定二次约束且目标函数为非凸二次函数的最优化问题提出了一类新的确定型全局优化算法,通过对目标函数和约束函数的线性下界估计,建立了原规划的松弛线性规划,通过对松弛线性规划可行域的细分以及一系列松弛线性规划的求解过程,得到原问题的全局最优解.我们从理论上证明了算法能收敛到原问题的全局最优解.     

求解非凸区域上凸函数比式和问题的全局优化方法

针对非凸区域上的凸函数比式和问题,给出一种求其全局最优解的确定性方法.该方法基于分支定界框架.首先通过引入变量,将原问题等价转化为d.c.规划问题,然后利用次梯度和凸包络构造松弛线性规划问题,从而将关键的估计下界问题转化为一系列线性规划问题,这些线性规划易于求解而且规模不变,更容易编程实现和应用到实际中;分支采用单纯形对分不但保证其穷举性,而且使得线性规划规模更小.理论分析和数值实验表明所提出的算法可行有效.     

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.     

Global optimization method for maximizing the sum of difference of convex functions ratios over nonconvex region

A global optimization algorithm is presented for maximizing the sum of difference of convex functions ratios problem over nonconvex feasible region. This algorithm is based on branch and bound framework. To obtain a difference of convex programming, the considered problem is first reformulated by introducing new variables as few as possible. By using subgradient and convex envelope, the fundamental problem of estimating lower bound in the branch and bound algorithm is transformed into a relaxed linear programming problem which can be solved efficiently. Furthermore, the size of the relaxed linear programming problem does not change during the algorithm search. Lastly, the convergence of the algorithm is analyzed and the numerical results are reported.     

A new global optimization approach for convex multiplicative programming

In this paper, by solving the relaxed quasiconcave programming problem in outcome space, a new global optimization algorithm for convex multiplicative programming is presented. Two kinds of techniques are employed to establish the algorithm. The first one is outer approximation technique which is applied to shrink relaxation area of quasiconcave programming problem and to compute appropriate feasible points and to raise the capacity of bounding. And the other one is branch and bound technique which is used to guarantee global optimization. Some numerical results are presented to demonstrate the effectiveness and feasibility of the proposed algorithm.     

A Global Optimization Method,QBB, for Twice-Differentiable Nonconvex Optimization Problem

A global optimization method, QBB, for twice-differentiable NLPs (Non-Linear Programming) is developed to operate within a branch-and-bound framework and require the construction of a relaxed convex problem on the basis of the quadratic lower bounding functions for the generic nonconvex structures. Within an exhaustive simplicial division of the constrained region, the rigorous quadratic underestimation function is constructed for the generic nonconvex function structure by virtue of the maximal eigenvalue analysis of the interval Hessian matrix. Each valid lower bound of the NLP problem with the division progress is computed by the convex programming of the relaxed optimization problem obtained by preserving the convex or linear terms, replacing the concave term with linear convex envelope, underestimating the special terms and the generic terms by using their customized tight convex lower bounding functions or the valid quadratic lower bounding functions, respectively. The standard convergence properties of the QBB algorithm for nonconvex global optimization problems are guaranteed. The preliminary computation studies are presented in order to evaluate the algorithmic efficiency of the proposed QBB approach.     

求解随机凸规划概率约束问题的对偶算法

1引言随机规划中的概率约束问题在工程和管理中有广泛的应用.因为问题中包含非线性的概率约束,它们的求解非常困难.如果目标函数是线性的,问题的求解就比较容易.给出了一个求解随机线性规划概率约束问题的综述.原-对偶算法和切平面算法是比较有效的.在本文中,我们讨论随机凸规划概率约束问题:     

A branch and bound network approach to the canonical constrained entropy problem

In this paper, we present a branch and bound algorithm for solving the constrained entropy mathematical programming problem. Unlike other methods for solving this problem, our method solves more general problems with inequality constraints. The advantage of the proposed technique is that the relaxed problem solved at each node is a singly constrained network problem. The disadvantage is that the relaxed problem has twice as many variables as the original problem. An application to regional planning is given, and an example problem is solved.     

A global optimization algorithm for linear fractional and bilinear programs

In this paper a deterministic method is proposed for the global optimization of mathematical programs that involve the sum of linear fractional and/or bilinear terms. Linear and nonlinear convex estimator functions are developed for the linear fractional and bilinear terms. Conditions under which these functions are nonredundant are established. It is shown that additional estimators can be obtained through projections of the feasible region that can also be incorporated in a convex nonlinear underestimator problem for predicting lower bounds for the global optimum. The proposed algorithm consists of a spatial branch and bound search for which several branching rules are discussed. Illustrative examples and computational results are presented to demonstrate the efficiency of the proposed algorithm.