一类改进的非凸二次规划有效集方法

一类改进的非凸二次规划有效集方法修乃华（河北师范学院数学系）ＡＣＬＡＳＳＯＦＩＭＰＲＯＶＥＤＡＣＴＩＶＥＳＥＴＭＥＴＨＯＤＳＦＯＲＮＯＮＣＯＮＶＥＸＱＵＡＤＲＡＴＩＣＰＲＯＧＲＡＭＭＩＮＧＰＲＯＢＬＥＭ￥ＸｉｕＮａｉ－ｈｕａ（Ｄｅｐｔ．ｏｆＭａｔｈ．...     

求解凸二次规划问题的不可行内点算法

该文对一般的凸二次规划问题，给出了一个不可行内点算法，并证明了该算法经过犗（狀２犔）步迭代之后，要么得到问题的一个近似最优解，要么说明该问题在某个较大的区域内无解．     

改进共轭梯度法求解无约束二次凸规划问题

针对共轭梯度法求解无约束二次凸规划时,在构造共轭方向上的局限性,对共轭梯度法进行了改进.给出了构造共轭方向的新方法,利用数学归纳法对新方法进行了证明.同时还给出了改进共轭梯度法在应用时的基本计算过程,并对方法的收敛性进行了证明.通过实例求解,说明了在求解二次无约束凸规划时,该方法相比共轭梯度法具有一定的优势.     

求解凸二次规划问题的势下降内点算法

1　引　言二次规划问题的求解是数学规划和工业应用等领域的一个重要课题 ,同时也是解一般非线性规划问题的序列二次规划算法的关键 .求解二次规划问题的早期技术是利用线性规划问题的单纯形方法求解二次规划问题的 KKT最优性必要条件[1 ] .这类算法比较直观 ,但在处理不等式约束时 ,松弛变量的引进很容易导致求解过程的明显减慢 .有效集策略是求解二次规划问题的另一类主要技术 .这类方法一般都是稳定的 ,但随着问题中大量不等式约束的出现 ,其收敛速度将越来越低[2 ] .简约空间技术将所求问题的 Hessian阵投影到自由变量所在的子空间中 …     

凸二次交叉规划的等价形式

利用参数规划逆问题考虑凸二次交叉规划与多目标规划的关系 ,把交叉规划转变为同变量规划组 ,再把同变量规划组变为多目标规划 ,证明了凸二次交叉规划的均衡解与多目标规划的最优解的关系。     

基于DC分解的非凸二次规划SDP近似解

本文提出一类基于DC分解的非凸二次规划问题SDP松弛方法,并通过求解一个二阶锥问题得到原问题的近似最优解.我们首先对非凸二次目标函数进行DC分解,然后利用线性下逼近得到一个凸二次松弛问题,而最优的DC分解可通过求解一个SDP问题得到.数值试验表明,基于DC分解的SDP近似解平均优于经典SDP松弛和随机化方法产生的近似解。     

求非凸二次约束二次规划问题全局解的线性化方法

1引言 考虑如下非凸二次规划的全局优化问题: (QP):{min xTQox doTx,s.t.xTQix ditx≤bi,i=1,…,m,x∈S={x∈Rn:l≤x≤u}, 其中Qo,Qi是n阶实对称矩阵,do,di∈Rn,bi∈R,i=1,…,m;l=(l1,…,ln)T,u=(u1,…,un)T .     

关于宏观经济的凸二次规划模型

本文运用凸二次规划理论，通过引入价格变量建立观经济的凸二次规划模型．并指出在社会主义市场经济中企业、产业、国家可以同时达到最优．     

凸二次规划的预估校正光滑算法

本文研究了凸二次规划的一种光滑算法,将规划的对应中心线条件改造成一个非线性方程组,对其应用牛顿法及其变形形式,并且证明了算法的全局收敛性.     

A neural network for solving a convex quadratic bilevel programming problem

A neural network is proposed for solving a convex quadratic bilevel programming problem. Based on Lyapunov and LaSalle theories, we prove strictly an important theoretical result that, for an arbitrary initial point, the trajectory of the proposed network does converge to the equilibrium, which corresponds to the optimal solution of a convex quadratic bilevel programming problem. Numerical simulation results show that the proposed neural network is feasible and efficient for a convex quadratic bilevel programming problem.     

Generalization of Murty's direct algorithm to linear and convex quadratic programming

Murty's algorithm for the linear complementarity problem is generalized to solve the optimality conditions for linear and convex quadratic programming problems with both equality and inequality constraints. An implementation is suggested which provides both efficiency and tight error control. Numerical experiments as well as field tests in various applications show favorable results.The author thanks K. G. Murty for his encouragement and helpful comments.     

Quadratic convex reformulations for quadratic 0–1 programming

This is a summary of the author’s PhD thesis supervised by A. Billionnet and S. Elloumi and defended on November 2006 at the CNAM, Paris (Conservatoire National des Arts et Métiers). The thesis is written in French and is available from http://www.cedric.cnam.fr/PUBLIS/RC1115. This work deals with exact solution methods based on reformulations for quadratic 0–1 programs under linear constraints. These problems are generally not convex; more precisely, the associated continuous relaxation is not a convex problem. We developed approaches with the aim of making the initial problem convex and of obtaining a good lower bound by continuous relaxation. The main contribution is a general method (called QCR) that we implemented and applied to classical combinatorial optimization problems.      

Dual support method for solving convex quadratic programs

In this article, we present a new dual method for solving convex (but not strictly convex) quadratic programs (QPs). Our method is the generalization of the dual support method, developed by Gabasov and co-workers in 1981, for solving convex QPs. It proceeds in two phases: the first is to construct the initial support, called coordinator support, for the problem and the second is to achieve the optimality of the problem. Results of numerical experiments are given comparing our approach with the active-set method.     

A new bound for the quadratic assignment problem based on convex quadratic programming

We describe a new convex quadratic programming bound for the quadratic assignment problem (QAP). The construction of the bound uses a semidefinite programming representation of a basic eigenvalue bound for QAP. The new bound dominates the well-known projected eigenvalue bound, and appears to be competitive with existing bounds in the trade-off between bound quality and computational effort. Received: February 2000 / Accepted: November 2000?Published online January 17, 2001     

Some properties of generalized proximal point methods for quadratic and linear programming

The proximal point method for convex optimization has been extended recently through the use of generalized distances (e.g., Bregman distances) instead of the Euclidean one. One advantage of these extensions is the possibility of eliminating certain constraints (mainly positivity) from the subproblems, transforming an inequality constrained problem into a sequence of unconstrained or equality constrained problems. We consider here methods obtained using a certain class of Bregman functions applied to convex quadratic (including linear) programming, which are of the interior-point type. We prove that the limit of the sequence generated by the method lies in the relative interior of the solution set, and furthermore is the closest optimal solution to the initial point, in the sense of the Bregman distance. These results do not hold for the standard proximal point method, i.e., when the square of the Euclidean norm is used as the Bregman distance.The research leading to this paper was partially supported by CNPq Grant 301280/86.We thank two anonymous referees whose comments and suggestions allowed us to improve our results in a significant way.     

An extension of Karmarkar's projective algorithm for convex quadratic programming

We present an extension of Karmarkar's linear programming algorithm for solving a more general group of optimization problems: convex quadratic programs. This extension is based on the iterated application of the objective augmentation and the projective transformation, followed by optimization over an inscribing ellipsoid centered at the current solution. It creates a sequence of interior feasible points that converge to the optimal feasible solution in O(Ln) iterations; each iteration can be computed in O(Ln ³) arithmetic operations, wheren is the number of variables andL is the number of bits in the input. In this paper, we emphasize its convergence property, practical efficiency, and relation to the ellipsoid method.     

MATHEMATICAL ANALYSIS OF SOME NEURAL NETWORKS FOR SOLVING LINEAR AND QUADRATIC PROGRAMMING

ＺＨＡＮＧＸＩＡＮＧＳＵＮ（章祥荪）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，Ｂｅｉｊｉｎｇ１０００８０，Ｃｈｉｎａ）ＲｅｃｅｉｖｅｄＪｕｎｅ１８，１９９４．Ｔｈｉｓｗｏｒｋｉ...     

A clique algorithm for standard quadratic programming

A standard Quadratic Programming problem (StQP) consists in minimizing a (nonconvex) quadratic form over the standard simplex. For solving a StQP we present an exact and a heuristic algorithm, that are based on new theoretical results for quadratic and convex optimization problems. With these results a StQP is reduced to a constrained nonlinear minimum weight clique problem in an associated graph. Such a clique problem, which does not seem to have been studied before, is then solved with an exact and a heuristic algorithm. Some computational experience shows that our algorithms are able to solve StQP problems of at least one order of magnitude larger than those reported in the literature.