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A Survey on Some Recent Developments of Alternating Direction Method of Multipliers

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Journal of the Operations Research Society of China 2022年第1期10卷 1-52页

作者： De-Ren Han LMIB of the Ministry of Education School of Mathematical SciencesBeihang UniversityBeijing 100191China

Recently, alternating direction method of multipliers (ADMM) attracts much attentions from various fields and there are many variant versions tailored for differentmodels. Moreover, its theoretical studies such as rate of convergence and extensionsto nonconvex problems also achieve much progress. In this paper, we give a surveyon some recent developments of ADMM and its variants.

关键词： Alternating direction method of multipliers global convergence Rate of convergence Nonconvex optimization

ON PERIODIC DYNAMICAL SYSTEMS

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Chinese Annals of Mathematics,Series B 2004年第4期25卷 455-462页

作者： LUWENLIAN CHENTIANPING LaboratoryofNonlinearMathematicsScience InstituteofMathematicsFudanUniversityShanghai200433China

The authors investigate the existence and the global stability of periodic solution for dynamical systems with periodic interconnections, inputs and self-inhibitions. The model is very general, the conditions are quit... 详细信息

关键词： Neural network Dynamical systems Periodic solution global convergence

Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

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Science China Mathematics 2023年第10期66卷 2361-2384页

作者： Wei Liu Ziqing Xie Wenfan Yi Key Laboratory of Computing and Stochastic Mathematics(Ministry of Education) Hunan Normal UniversityChangsha 410081China School of Mathematics Hunan UniversityChangsha 410082China

The local minimax method(LMM)proposed by Li and Zhou(2001,2002)is an efficient method to solve nonlinear elliptic partial differential equations(PDEs)with certain variational structures for multiple *** steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou(2002)and play a significant role in the performance and convergence analysis of traditional *** this paper,a new algorithm framework of the LMMs is established based on general descent directions and two normalized(strong)Wolfe-Powell-type step-size search *** corresponding algorithm framework,named the normalized Wolfe-Powell-type LMM(NWP-LMM),is introduced with its feasibility and global convergence rigorously justified for general descent *** a special case,the global convergence of the NWP-LMM combined with the preconditioned steepest descent(PSD)directions is also ***,it extends the framework of traditional *** addition,conjugate-gradient-type(CG-type)descent directions are utilized to speed up the ***,extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared with different algorithms in the LMM’s family to indicate the effectiveness and robustness of our *** practice,the NWP-LMM combined with the CG-type direction performs much better than its known LMM companions.

关键词： semilinear elliptic PDE multiple unstable solution local minimax method normalized strong Wolfe-Powell-type search rule conjugate-gradient-type descent direction general descent direction global convergence

AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS

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Journal of Computational Mathematics 2021年第3期39卷 358-374页

作者： Yuting Chen Mingyuan Cao Yueting Yang Qingdao Huang School of Mathematics Jilin UniversityChangchun 130012China School of Mathematics and Statistics Beihua UniversityJilin 132013China

For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit *** this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric *** of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm *** other one is that a projection scheme is used to ensure the feasibility of all *** convergence and local quadratic convergence of our algorithm are established,*** preliminary numerical results show the efficiency of the proposed algorithm.

关键词： Symmetric tensors Generalized eigenvalues Trust-region global convergence Local quadratic convergence

Further insight into the convergence of the Fletcher-Reeves method

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Science China Mathematics 1999年第9期42卷 905-916页

作者：戴彧虹 State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Chinese Academy of Sciences Beijing China

The convergence properties of the Fletcher-Reeves method for unconstrained optimization are further studied with the technique of generalized line search. Two conditions are given which guarantee the global convergence of the Fletcher-Reeves method using generalized Wolfe line searches or generalized Arjimo line searches, whereas an example is constructed showing that the conditions cannot be relaxed in certain senses.

关键词： unconstrained optimization conjugate gradient Fletcher-Reeves method generalid line search global convergence

A Barzilai-Borwein conjugate gradient method

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Science China Mathematics 2016年第8期59卷 1511-1524页

作者： DAI YuHong KOU CaiXia LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science Chinese Academy of Sciences School of Science Beijing University of Posts and Telecommunications

The linear conjugate gradient method is an optimal method for convex quadratic minimization due to the Krylov subspace minimization property. The proposition of limited-memory BFGS method and Barzilai-Borwein gradient method, however, heavily restricted the use of conjugate gradient method for largescale nonlinear optimization. This is, to the great extent, due to the requirement of a relatively exact line search at each iteration and the loss of conjugacy property of the search directions in various occasions. On the contrary, the limited-memory BFGS method and the Barzilai-Bowein gradient method share the so-called asymptotical one stepsize per line-search property, namely, the trial stepsize in the method will asymptotically be accepted by the line search when the iteration is close to the solution. This paper will focus on the analysis of the subspace minimization conjugate gradient method by Yuan and Stoer(1995). Specifically, if choosing the parameter in the method by combining the Barzilai-Borwein idea, we will be able to provide some efficient Barzilai-Borwein conjugate gradient(BBCG) methods. The initial numerical experiments show that one of the variants, BBCG3, is specially efficient among many others without line searches. This variant of the BBCG method might enjoy the asymptotical one stepsize per line-search property and become a strong candidate for large-scale nonlinear optimization.

关键词： conjugate gradient method subspace minimization Barzilai-Bowein gradient method line search descent property global convergence

A novel evolutionary algorithm for global numerical optimization with continuous variables

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Progress in Natural Science:Materials International 2008年第3期18卷 345-351页

作者： Wenhong Zhaoa,b, Wei Wang a, Yuping Wang a, a School of Computer Science and Technology, Xidian University, Xi’an 710071, China b Faculty of Science, Xidian University, Xi’an 710071, China School of Computer Science and Technology Xidian University Xi’an 710071 China Faculty of Science Xidian University Xi’an 710071 China

Evolutionary algorithms (EAs) are a class of general optimization algorithms which are applicable to functions that are multimodal, non-differentiable, or even discontinuous. In this paper, a novel evolutionary algorithm is proposed to solve global numerical optimiza- tion with continuous variables. In order to make the algorithm more robust, the initial population is generated by combining determinate factors with random ones, and a decent scale function is designed to tailor the crossover operator so that it can not only find the decent direction quickly but also keep scanning evenly in the whole feasible space. In addition, to improve the performance of the algorithm, a mutation operator which increases the convergence-rate and ensures the convergence of the proposed algorithm is designed. Then, the global convergence of the presented algorithm is proved in detail. Finally, the presented algorithm is executed to solve 24 benchmark problems, and the results show that the convergence-rate of the proposed algorithm is much faster than that of the compared algorithms.

关键词： Evolutionary algorithm Decent scale function global numerical optimization global convergence

An SQP algorithm for mathematical programs with nonlinear complementarity constraints

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Applied Mathematics and Mechanics(English Edition) 2009年第5期30卷 659-668页

作者：朱志斌简金宝张聪 School of Mathematics & Computational Science Guilin University of Electronic Technology College of Mathematics and Information Science Guangxi University

In this paper, we describe a successive approximation and smooth sequential quadratic programming （SQP） method for mathematical programs with nonlinear complementarity constraints （MPCC）. We introduce a class of smooth programs to approximate the MPCC. Using an 11 penalty function, the line search assures global convergence, while the superlinear convergence rate is shown under the strictly complementary and second-order sufficient conditions. Moreover, we prove that the current iterated point is an exact stationary point of the mathematical programs with equilibrium constraints （MPEC） when the algorithm terminates finitely.

关键词： mathematical programs with equilibrium constraints (MPEC) SQP algorithm successive approximation global convergence superlinear convergence rate

A Nonmonotone Line Search Based Algorithm for Distribution Center Location Selected

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Acta Mathematicae Applicatae Sinica 2014年第3期30卷 699-706页

作者： Zhu-cui JING Meng-gang LI Chuan-long WANG School of Economics and Management Beijing Jiaotong University CCISR Beijing Jiaotong University Department of Mathematics Taiyuan Teachers' College

The minimax optimization model introduced in this paper is an important model which has received some attention over the past years. In this paper, the application of minimax model on how to select the distribution center location is first introduced. Then a new algorithm with nonmonotone line search to solve the non-decomposable minimax optimization is proposed. We prove that the new algorithm is global Convergent. Numerical results show the proposed algorithm is effective.

关键词： non-decomposable minimax nonmonotone line search global convergence