不等式约束优化基于新型积极识别集的SQCQP算法

本文提出一个新的求解非线性不等式约束优化问题的罚函数型序列二次约束二次规划(SQCQP)算法.算法每次迭代只需求解一个凸二次约束二次规划(QCQP)子问题,且通过引入新型积极识别集技术,QCQP子问题的规模显著减小,从而降低计算成本.在不需要函数凸性等较弱假设下,算法具有全局收敛性.初步的数值试验表明算法是稳定有效的.     

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。     

连续化方法求解约束凸规划问题

1．引言大型规划问题数值求解一直是计算数学工作者感兴趣的课题之一．针对大型约束规划问题,1991年李兴斯山提出凝聚函数法,该方法用光滑的凝聚函数逼近非光滑的极大值函数,从而把多个约束函数转化为带参数的单个光滑函数约束,从而降低了问题的规模．近年来,K3］研究了凸规划问题的凝聚函数法的收敛性,在目标函数强凸性及对一般凸规划研究了收敛性质．向讨论了可行解集有界的线性规划问题的凝聚函数求解算法并证明了收效性定理．上述文章均预先把凝聚参数取得充分小,然后对固定参数的单约束近似问题进行求解．一般地,凝聚参数取得…     

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:     

一类单调非凸约束最优规划修正的新型分枝定界算法

本文讨论了一类单调非凸约束最优规划的目标函数和约束集的结构特征性质.阐明了如何将所考虑的问题等价地转化为一个递增函数在另一个递增函数水平集上的极大优化问题.在此基础上提出了一个我们称之为修正的新型分枝定界算法.新算法的修正之处是在计算新的极点时,采用了一个有效的新的区域删除模式以构造越来越小的Polyblock集覆盖EnH且不舍y,以排除问题(P)可行域中不存在全局r最优解的部分.最后,证明了算法的收敛性.初步的数值实验表明算法是有效可行的,可应用于求解更广的一类非凸最优规划.     

非凸约束优化的随机近似算法

在人工智能、科学计算等领域,众多应用驱动的数学优化模型因依赖于庞大的数据集和/或不确定的信息而呈现出随机性、且伴有复杂非凸算子约束。于是精确计算模型中的函数信息往往代价高昂,同时非凸约束的存在也给模型求解和算法分析带来极大的挑战。近年来,结合模型的结构、利用函数的随机近似信息来设计、分析非凸约束优化算法开始引起关注。目前主流的求解非凸约束优化的随机近似算法主要分为三类:基于随机近似的罚方法、邻近点算法和随机序列二次规划算法。本文对这几类算法的研究进展进行梳理和总结,简要地介绍相关算法的设计思想和基本的理论性质,如渐近收敛性理论、复杂度理论等。     

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.     

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.     

基于最优D.C.分解的单二次约束非凸二次规划精确算法

本文提出一种基于最优D.C.分解的单二次约束非凸二次规划精确算法.本文首先对非凸二次日标函数进行D.C.分解,然后对D.C.分解中凹的部分进行线性下逼近得到一个凸二次松弛问题.本文证明了最优D.C.分解可通过求解一个半定规划问题得到,而原问题的最优解可以通过计算最优凸二次松弛问题的满足某种互补条件的解得到.最后,本文报告了初步数值计算结果.     

一类光滑凸规划的牛顿法

给出了一个求解一类光滑凸规划的算法,利用光滑精确乘子罚函数把一个光滑凸规划的极小化问题化为一个紧集上强凸函数的极小化问题,然后在给定的紧集上用牛顿法对这个强凸函数进行极小化．     

线性规划新算法的改进

本文基于算法要简单实用的思想,对一种线性规划新算法中的核心算法进行改进,使其计算方法更简单计算量更小,使整个算法更为可行有效。     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

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.     

A parametric successive underestimation method for convex multiplicative programming problems

This paper addresses itself to the algorithm for minimizing the product of two nonnegative convex functions over a convex set. It is shown that the global minimum of this nonconvex problem can be obtained by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in a higher dimensional space and to apply a branch-and-bound algorithm using an underestimating function. Computational results indicate that our algorithm is efficient when the objective function is the product of a linear and a quadratic functions and the constraints are linear. An extension of our algorithm for minimizing the sum of a convex function and a product of two convex functions is also discussed.     

A very simple SQCQP method for a class of smooth convex constrained minimization problems with nice convergence results

We introduce a new and very simple algorithm for a class of smooth convex constrained minimization problems which is an iterative scheme related to sequential quadratically constrained quadratic programming methods, called sequential simple quadratic method (SSQM). The computational simplicity of SSQM, which uses first-order information, makes it suitable for large scale problems. Theoretical results under standard assumptions are given proving that the whole sequence built by the algorithm converges to a solution and becomes feasible after a finite number of iterations. When in addition the objective function is strongly convex then asymptotic linear rate of convergence is established.     

Piecewise-linear programming: The compact (CPLP) algorithm

A compact algorithm is presented for solving the convex piecewise-linear-programming problem, formulated by means of a separable convex piecewise-linear objective function (to be minimized) and a set of linear constraints. This algorithm consists of a finite sequence of cycles, derived from the simplex method, characteritic of linear programming, and the line search, characteristic of nonlinear programming. Both the required storage and amount of calculation are reduced with respect to the usual approach, based on a linear-programming formulation with an expanded tableau. The tableau dimensions arem×(n+1), wherem is the number of constraints andn the number of the (original) structural variables, and they do not increase with the number of breakpoints of the piecewise-linear terms constituting the objective function.     

Simplex-inspired algorithms for solving a class of convex programming problems

We consider a class of convex programming problems whose objective function is given as a linear function plus a convex function whose arguments are linear functions of the decision variables and whose feasible region is a polytope. We show that there exists an optimal solution to this class of problems on a face of the constraint polytope of dimension not more than the number of arguments of the convex function. Based on this result, we develop a method to solve this problem that is inspired by the simplex method for linear programming. It is shown that this method terminates in a finite number of iterations in the special case that the convex function has only a single argument. We then use this insight to develop a second algorithm that solves the problem in a finite number of iterations for an arbitrary number of arguments in the convex function. A computational study illustrates the efficiency of the algorithm and suggests that the average-case performance of these algorithms is a polynomial of low order in the number of decision variables. The work of T. C. Sharkey was supported by a National Science Foundation Graduate Research Fellowship. The work of H. E. Romeijn was supported by the National Science Foundation under Grant No. DMI-0355533.     

Convergence property of the Iri-Imai algorithm for some smooth convex programming problems

In this paper, the Iri-Imai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions satisfy a certain type of convexity, called the harmonic convexity in this paper. A characterization of this convexity condition is given. The same convexity condition was used by Mehrotra and Sun to prove the convergence of a path-following algorithm.The Iri-Imai algorithm is a natural generalization of the original Newton algorithm to constrained convex programming. Other known convergent interior-point algorithms for smooth convex programming are mainly based on the path-following approach.     

An algorithm for nonconvex programming problems

Branch and bound approaches for nonconvex programming problems had been given in [1] and [4]. Crucial for both are the use of rectangular partitions, convex envelopes and separable nonconvex portions of the objective function and constraints. We want to propose a similar algorithm which solves a sequence of problems in each of which the objective function is convex or even linear. The main difference between this approach and previous approaches is the use of general compact partitions instead of rectangular ones and a different refining rule such that the algorithm does not rely on the concept of convex envelopes and handles non-separable functions.First we describe a general algorithm and prove a convergence theorem under suitable regularity assumptions. Then we give as example an algorithm for concave minimization problems.     

Using convex envelopes to solve the interactive fixed-charge linear programming problem

In the well-known fixed-charge linear programming problem, it is assumed, for each activity, that the value of the fixed charge incurred when the level of the activity is positive does not depend upon which other activities, if any, are also undertaken at a positive level. However, we have encountered several practical problems where this assumption does not hold. In an earlier paper, we developed a new problem, called the interactive fixed-charge linear programming problem (IFCLP), to model these problems. In this paper, we show how to construct the convex envelopes and other convex underestimating functions for the objective function for problem (IFCLP) over various rectangular subsets of its domain. Using these results, we develop a specialized branch-and-bound algorithm for problem (IFCLP) which finds an exact optimal solution for the problem in a finite number of steps. We also discuss the main properties of this algorithm.The authors would like to thank an anonymous referee for his helpful suggestions.