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Constrained Optimization: Projected Gradient Flows

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2009年第1期140卷 117-130页

作者： Shikhman, V. Stein, O. Rhein Westfal TH Aachen Dept Math C D-52056 Aachen Germany Univ Karlsruhe TH Sch Econ & Business Engn D-76128 Karlsruhe Germany

We consider a dynamical system approach to solve finite-dimensional smooth optimization problems with a compact and connected feasible set. In fact, by the well-known technique of equalizing inequality constraints using quadratic slack variables, we transform a general optimization problem into an associated problem without inequality constraints in a higher-dimensional space. We compute the projected gradient for the latter problem and consider its projection on the feasible set in the original, lower-dimensional space. In this way, we obtain an ordinary differential equation in the original variables, which is specially adapted to treat inequality constraints (for the idea, see Jongen and Stein, Frontiers in Global Optimization, pp. 223-236, Kluwer Academic, Dordrecht, 2003). The article shows that the derived ordinary differential equation possesses the basic properties which make it appropriate to solve the underlying optimization problem: the longtime behavior of its trajectories becomes stationary, all singularities are critical points, and the stable singularities are exactly the local minima. Finally, we sketch two numerical methods based on our approach.

关键词： Constrained optimization variable metric algorithms Quadratic slack variables Projected gradient algorithms Gradient systems

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A CLASS OF REVISED BROYDEN algorithms WITHOUT EXACT LINE SEARCH

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Journal of Computational Mathematics 2004年第1期22卷 11-20页

作者： Ding-guoPu Sheng-huaGui Wei-wenTian DepartmentofMathematics ShanghaiTongjiUniversityShanghai200331China

In this paper, we discuss the convergence of the Broyden algorithms with revised search direction. Under some inexact line searches, we prove that the algorithms are globally convergent for continuously differentiable functions and the rate of local convergence of the algorithms is one-step superlinear and n-step second-order for uniformly convex objective functions.

关键词： Convergence Convergence rate Line search variable metric algorithms

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A CLASS OF REVISED BROYDEN algorithms

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Journal of Systems Science & Complexity 2003年第2期16卷 213-221页

作者： PUDingguo TIANWeiwen DepartmentofAppliedMathematics TongjiUniversityShanghai200333China ShanghaiUniversity Shanghai200436China

In this paper, we discuss the convergence of the Broyden algorithms withrevised search direction. Under some inexact line searches, we prove that the algorithms areglobally convergent for continuously differentiable functions and the rate of convergence of thealgorithms is one-step superlinear and n-step second-order for uniformly convex objective functions.

关键词： variable metric algorithms line search convergence convergence rate

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《动物医学进展》参考文献著录规则

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动物医学进展 2001年第1期22卷 96-96页

1.参考文献文内标著格式　　采用“顺序编码制”，在引文处，对引用的文献，按它们在论著中出现的先后用阿拉伯数字连续排序，将序号置于方括号内，并视具体情况，把序号做为上角标，或者做为语句的组成部分。　　2.文后参考文献编写格... 详细信息

1.参考文献文内标著格式　　采用“顺序编码制”，在引文处，对引用的文献，按它们在论著中出现的先后用阿拉伯数字连续排序，将序号置于方括号内，并视具体情况，把序号做为上角标，或者做为语句的组成部分。　　2.文后参考文献编写格式　　采用“顺序编码制”时，在文后参考文献表中，各条文献在论文中的文献序号顺序排列，项目完整(包括作者、文题、刊名或书名、出版年、出版地、卷、期、页次)，内容准确，各个项目的次序和著录符号应符合规定。3.文献著录的作者　　对于个人作者(包括译者、编者)，在参考文献表中，用各种文种式书写的姓名，一律姓在前名在后；外国人名可缩写为首字母(大写)，但不加缩写点(.)。作者为3人或少于3人应全部写出，之间用“，”号相隔，3人以上只列出前3人，后加“等”或相应的文字如“et al”，在“et al”之前加“，”号。4.各类文献的著录格式及示例　　a.书(专著)［序号］　作者.书名(版本)［M］.出版地：出版者，出版年.页次.　　示例：［2］　谭丙煜.怎样撰写科学论文［M］.沈阳：辽宁人民出版社，1982 ．59．b.期刊［序号］　作者.题名［J］.刊名，出版年,卷号(期号)：页次.示例：［1］　姚振兴，郑天愉，曹柏如，等.用P波波形资料测定中强地震震源过程的方法［J］.地球物理进展，1991，6(4)：34～36．［2］　Fletcher R. A new approach to variable metric algorithms［J］.Computer J,1970，13(2):317～322．［3］　贝拉斯.光纤陀螺仪样机的测试结果［J］.刘燕华译.舰船导航，1990，16( 4)：74．c.论文集［序号］　作者.题名［C］.出版地：出版者，出版年．页次．　　示例：［2］　辛希孟．信息技术与信息服务国际研讨会论文集：A集［C］．北京：中国社会科学出版社，1994．　　d.论文集中的析出文献［序号］　析出文献主要作者.析出文献题名［A］．原文献主要作者. 原文献题名［C］.出版地：出版者，出版年．页次．　　示例：［5］　赵秀珍.关于计算机学科中几个量和单位用法的建议［A］.中国高等学校自然科学学报研究会编.科技编辑学论文集［C］.北京：北京师范大学出版社，1997．125～129．　　e.学位论文［序号］　作者.题名［D］.出版地：出版者，出版年．页次．　　示例：［13］　陈淮金.多机电力系统分散最优励磁控制器的研究［D］.北京：清华大学电机工程系，1988．　　f.报告　［序号］　作者．题名［R］．出版地：出版者，出版年．页次．　　示例：［4］　冯西桥．核反应堆压力管道与压力容器的LBB分析［R］．北京：清华大学核能技术设计研究院，1997．　　g.专利文献［序号］　专利申请者.专利题名［P］.专利国别：专利号，出版日期．示例：［10］　姜锡洲.一种温热外敷药制备方法［P］.中国专利：881056073 ，1989-07-26．　　h.各种未定义类型的文献［序号］　作者．题名［Z］．出版地：出版者，出版年．

关键词：动物医学进展出版者文后参考文献表文献题名论文 variable metric algorithms 专利顺序编码制北京文集清华大学最优励磁控制器著录中国原文献析出格式多机电力系统地震震源过程

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On the convergence of the DFP algorithm for unconstrained optimization when there are only two variables

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MATHEMATICAL PROGRAMMING 2000年第2期87卷 281-301页

作者： Powell, MJD Univ Cambridge Dept Appl Math & Theoret Phys Cambridge CB3 9EW England

Let the DFP algorithm for unconstrained optimization be applied to an objective function that has continuous second derivatives and bounded level sets, where each line search finds the first local minimum. it is proved that the calculated gradients are not bounded away from zero if there are only two variables. The new feature of this work is that there is no need for the objective function to be convex.

关键词： convergence theory unconstrained optimization variable metric algorithms

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THE CONVERGENCE OF BROYDEN algorithms FOR LC GRADIENT FUNCTION

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Acta Mathematicae Applicatae Sinica 2000年第3期16卷 313-319页

作者：濮定国 Institute of Applied Mathematics Shanghai Tiedao University Shanghai China

In this paper, we discuss the convergence of Broyden algorithms for the functions which are non-twice differentiable, but have LC gradient. We prove that the rate of convergence of the algorithms is linear for uniformly convex functions. We also demonstrate that under some mild conditions the algorithms are superlinsarly convergent.

关键词： variable metric algorithms convex convergence convergence rate

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A class of Broyden algorithms with revised search directions

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ASIA-PACIFIC JOURNAL OF OPERATIONAL RESEARCH 1997年第2期14卷 93-109页

作者： Pu, DG Tian, WW City Univ Hong Kong Dept Math Hong Kong Peoples R China Shanghai Univ Dept Math Jiading Peoples R China

In this paper we discuss the convergence of the Broyden algorithms with revised search direction. We prove that the algorithms are globally convergent for continuously differentiable functions and the rate of converge... 详细信息

关键词： variable metric algorithms line search convergence convergence rate

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A quasi-second-order proximal bundle algorithm

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MATHEMATICAL PROGRAMMING 1996年第1期73卷 51-72页

作者： Mifflin, R Department of Pure and Applied Mathematics Washington State University Pullman USA

This paper introduces an algorithm for convex minimization which includes quasi-Newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm. The method uses the Hessian of a certain outer function which depends on the Jacobian of a proximal point mapping which, in turn, depends on the preconditioner matrix and on a Lagrangian Hessian relative to a certain tangent space. Convergence is proved under boundedness assumptions on the preconditioner sequence.

关键词： bundle methods convex minimization global convergence proximal point quasi-Newton variable metric algorithms

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MODIFYING THE BFGS UPDATE BY A NEW COLUMN SCALING TECHNIQUE

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MATHEMATICAL PROGRAMMING 1994年第1期66卷 45-78页

作者： SIEGEL, D 1. Department of Applied Mathematics and Theoretical Physics Silver Street Cambridge UK

Let B be a positive definite symmetric approximation to the second derivative matrix of the objective function of a minimization calculation. We study modifications of the BFGS method that apply a scaling technique to the columns of a conjugate direction matrix Z satisfying Z(T)BZ = I. For a simple scaling technique similar to the ones considered by Powell (1987) and (1989) we show that, due to a two iteration cycle, linear convergence can occur when the method is applied to a quadratic function and Wolfe's line search is employed, although for exact line searches quadratic termination can be proved. We then suggest a different scaling technique that prevents certain columns thought to contain important curvature information from being scaled. For this algorithm we prove global and superlinear convergence and demonstrate the superiority of our method over the BFGS formula on a range of ill-conditioned optimization problems. Moreover, we present an implementation of our algorithm tht requires only 3n2 + O(n) multiplications per iteration.

关键词： CONJUGATE DIRECTIONS MATRIX FACTORIZATIONS NONLINEAR OPTIMIZATION UPDATING variable metric algorithms

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LINE SEARCH algorithms WITH GUARANTEED SUFFICIENT DECREASE

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 1994年第3期20卷 286-307页

作者： MORE, JJ THUENTE, DJ INDIANA PURDUE UNIV DEPT MATH SCIFT WAYNEIN 46805

The development of software for minimization problems is often based on a line search method. We consider line search methods that satisfy sufficient decrease and curvature conditions, and formulate the problem of determining a point that satisfies these two conditions in terms of finding a point in a set T(mu). We describe a search algorithm for this problem that produces a sequence of iterates that converge to a point in T(mu) and that, except for pathological cases, terminates in a finite number of steps. Numerical results for an implementation of the search algorithm on a set of test functions show that the algorithm terminates within a small number of iterations.

关键词： algorithms CONJUGATE GRADIENT algorithms LINE SEARCH algorithms NONLINEAR OPTIMIZATION TRUNCATED NEWTON algorithms variable metric algorithms

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