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Block coordinate descent with time perturbation for nonconvex nonsmooth problems in real-world studies

Frontiers of Information Technology & Electronic Engineering 2019年第10期20卷 1390-1403页

作者： Rui LIU Wei-chu SUN Tao HOU Chun-hong HU Lin-bo QIAO Department of Oncology The Second Xiangya Hospital of Central South UniversityChangsha 410011China First Clinical Medical College University of South ChinaHengyang 421001China College of Computer National University of Defense TechnologyChangsha 410073China

The era of big data in healthcare is here, and this era will significantly improve medicine and especially oncology. However, traditional machine learning algorithms need to be promoted to solve such large-scale real-world problems due to a large amount of data that needs to be analyzed and the difficulty in solving problems with nonconvex nonlinear settings. We aim to minimize the composite of a smooth nonlinear function and a block-separable nonconvex function on a large number of block variables with inequality constraints. We propose a novel parallel first-order optimization method, called asynchronous block coordinate descent with time perturbation (ATP), which adopts a time perturbation technique that escapes from saddle points and sub-optimal local points. The details of the proposed method are presented with analyses of convergence and iteration complexity properties. Experiments conducted on real-world machine learning problems validate the efficacy of our proposed method. The experimental results demonstrate that time perturbation enables ATP to escape from saddle points and sub-optimal points, providing a promising way to handle nonconvex optimization problems with inequality constraints employing asynchronous block coordinate descent. The asynchronous parallel implementation on shared memory multi-core platforms indicates that the proposed algorithm, ATP, has strong scalability.

关键词： Convergence analysis Asynchronous block coordinate descent method Time perturbation Nonconvex nonsmooth optimization Real-world study

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A Class of Alternating Linearization Algorithms for nonsmooth Convex Optimization

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Acta Mathematicae Applicatae Sinica 2019年第2期35卷 435-443页

作者： Dan LI Jie SHEN Yuan LU Li-Ping PANG Zun-Quan XIA Information and Engineering College Dalian University School of Mathematics Liaoning Normal University Normal College Shenyang University School of Mathematical Sciences Dalian University of Technology

We consider the problems of minimizing the sum of a continuously differentiable convex function and a nonsmooth convex function in this paper. These problems arise in many applications of practical interest.A class of alternating linearization methods is presented for solving these problems. The global convergence rate is also obtained under certain mild conditions. Numerical experiments validate the theoretical convergence analysis and verify the implementation of the proposed algorithm.

关键词： alternating linearization method proximal point convex programming nonsmooth optimization

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TRACES OF nonsmooth FUNCTIONS ON POLYHEDRAL DOMAINS

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Science China Mathematics 1992年第4期35卷 452-462页

作者：丁桦 Institute of Mechanics Academia Sinica Beijing 100080 PRC

Introduced in this paper are the definitions of the traces for a class of nonsmooth func-tions on polyhedral domains. By analyzing their properties we get the structures of thcsetraces.

关键词： traces polyhedral domain nonsmooth function.

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Nonconforming Finite Element Methods for the Constrained Optimal Control Problems Governed by nonsmooth Elliptic Equations

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Acta Mathematicae Applicatae Sinica 2020年第2期36卷 471-481页

作者： Hong-bo GUAN Dong-yang SHI College of Mathematics and Information Sciences Zhengzhou University of Light IndustryZhengzhou 450002China School of Mathematics and Statistics Zhengzhou UniversityZhengzhou 450001China

In this paper,nonconforming finite element methods(FEMs)are proposed for the constrained optimal control problems(OCPs)governed by the nonsmooth elliptic equations,in which the popular EQr1 ot element is employed to approximate the state and adjoint state,and the piecewise constant element is used to approximate the control.Firstly,the convergence and superconvergence properties for the nonsmooth elliptic equation are obtained by introducing an auxiliary problem.Secondly,the goal-oriented error estimates are obtained for the objective function through establishing the negative norm error estimate.Lastly,the methods are extended to some other well-known nonconforming elements.

关键词： nonconforming finite element supercloseness and superconvergence optimal control problems nonsmooth elliptic equations goal-oriented error estimate

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nonsmooth finite-time control of uncertain second-order nonlinear systems

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控制理论与应用（英文版） 2007年第2期5卷 171-176页

作者： Shuanghe YU Zi MA Xiaohui YANG Automation Research Center Dalian Maritime UniversityDalian Liaoning 116026 China

nonsmooth finite-time stabilizing control laws have been developed for the double integrator system. The objective of this paper is to further explore the finite-time tracking control problem of a general form of uncertain secondorder affine nonlinear system with the new forms of terminal sliding mode （TSM）. Discontinuous and continuous finite-time controllers are also developed respectively without the singularity problem. Complete robustness can be acquired with the former, and enhanced robustness compared with the conventional boundary layer method can be expressed as explicit bounded function with the latter. Simulation results on the stabilizing and tracking problems are presented to demonstrate the effectiveness of the control algorithms.

关键词： nonsmooth Finite-time stability Terminal sliding mode

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A nonmonotone trust region algorithm for unconstrained nonsmooth optimization

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Chinese Science Bulletin 1996年第3期41卷 197-201页

作者：柯小伍刘光辉徐大川 Department of Mathematics Beijing Normal University Beijing 100875 China Institute of Applied Mathematics Chinese Academy of Sciences Beijing 100080 China

In this note, the following unconstrained nonsmooth optimization problem is considered where f(x):R^n→R is only a locally Lipschitzian function. Many papers appear on the convergence properties of the trust region algorithm to solve several different particular nonsmooth problems. Dennis, Li and Tapia proposed a general trust region model by using regular functions. They proved the global convergence of the general trust region model under some mild conditions which are shown to be satisfied by many trust region algorithms including smooth one. Qi and Sun provided another trust region model

关键词： trust region algorithms locally Lipschitzian functions global convergence nonmonotone nonsmooth optimization.

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nonsmooth MULTIOBJECTIVE PROGRAMMING

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Systems Science and Mathematical Sciences 1994年第4期7卷 362-366页

作者： WANG Liancheng DONG Jiali LIU Qinghuai(Mathematics Department, Jilin University of Technology, Changchun 130025,China) 吉林工业大学数学系吉林长春

（ｘ）？Ｓ．Ｉｆｇ（ｘ）＝０，Ｓ＊（ｘ）＝Ｓ＊．ＭｏｍｎｉｔＳ／０ａｎｄｎｉｔＳ（ｘ）２ｎｉｔＳ，ｎｉｔＳ（ｘ）／０．Ｈｅｌｌｃｅ（Ｓ（ｘ））＊＝Ｓ＊（ｘ）ｈａｓａｃｏｍｐａｃｔｃｏｎｖｅｘｂａｓｅ，ｄｅｎｏｔ... 详细信息

（ｘ）？Ｓ．Ｉｆｇ（ｘ）＝０，Ｓ＊（ｘ）＝Ｓ＊．ＭｏｍｎｉｔＳ／０ａｎｄｎｉｔＳ（ｘ）２ｎｉｔＳ，ｎｉｔＳ（ｘ）／０．Ｈｅｌｌｃｅ（Ｓ（ｘ））＊＝Ｓ＊（ｘ）ｈａｓａｃｏｍｐａｃｔｃｏｎｖｅｘｂａｓｅ，ｄｅｎｏｔｅｄｂｙＡ，ａｎｄｔｈｅｃｏｍｐａｃｔｃｏｎｖｅｘｂａｓｅｏｆＱ＊ｉｓｄｅｎｏｔｅｄｂｙＢ．｛０｝＊＝Ｒｅ．ＬｅｔＣ＝｛ｘＥＲｅ：ＩＩｘｌｌ＝１｝．ＴｈｅｎＣｉｓａｃｏｍｐａｃｔｂａｓｅｏｆ｛０｝＊．Ｌｅｔｓｕｐｐｏｓｅｔｈａｔ（３ｄＯＥＲ＂）（ＶＰＥＶ＊＼｛０｝）（ＰＴｐ）＂（ｘ；ｄｏ）＜０，ｆｒｏｍＬｅｍｍａ２．３ｔｈｅｒｅｅｄｅｓｔｓａｄＥＲ＂ｔｏｇｅｔｈｅｒｗｉｔｈｂ＞０ｓｕｃｈｔｈａｔＩｎｐａｒｔｉｃｕｌａｒ，ｆｉｘｉｎｇＴＥＢａｎｄＡ６Ａ，ｓｅｔｔｉｎｇｐ￣（ＥＴ，ａｃ，￣）ＥＶ＊ｗｉｔｈＥ＞０ａｎｄｕｅＣ，ｗｅＩｎｄｅｅｄ，ｉｆｔｈｅｒｅｅｘｉｓｔｓａａｌＥ（０，６）ｓｕｃｈｔｈａｔｈ（ｘ＋ａ＇ｄ）／０，ｗｅｃａｎｃｈｏｏｓｅＦＩＥＣｓａｔｉｓｆｙｉｎｇＵ，＂ｈ（ｘ＋ａ＇ｄ）＞０．Ｉｎ（４），ｌｅｔｔｉｎｇＥ－０，ｉｔｆｏｌｌｏｗｓｔｈａｔＵ＂ｈ（ｘ＋ｒｓ＇ｄ）＜０，ｗｈ?

关键词： nonsmooth multiobjective programming weak minimum necessary and sufficient condition

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nonsmooth Optimization Algorithms in Some Problems of Fracture Dynamics

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Intelligent Information Management 2010年第11期2卷 637-646页

作者： V. V. Zozulya 不详

Mathematical statement of elastodynamic contact problem for cracked body with considering unilateral restrictions and friction of the crack faces is done in classical and weak forms. Different variational formulations of unilateral contact problems with friction based on boundary variational principle are considered. nonsmooth optimization algorithms of Udzawa’s type for solution of unilateral contact problem with friction have been developed. Convergence of the proposed algorithms has been studied numerically.

关键词： Unilateral Contact Friction Crack Variational Principles Boundary Variational Functional nonsmooth Optimization Algorithm

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NONDESCENT SUBGRADIENT METHOD FOR nonsmooth CONSTRAINED MINIMIZATION

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1994年第2期3卷 126-134页

作者：徐慧福 Department of Mathematics Ningbo University Ningbo 315211 Zhejiang Province PRC.

A kind of nondecreasing subgradient algorithm with appropriate stopping rule has been proposed for nonsmooth constrained minimization problem. The dual theory is invoked in dealing with the stopping rule and general global minimiizing algorithm is employed as a subroutine of the algorithm. The method is expected to tackle a large class of nonsmooth constrained minimization problem.

关键词： nonsmooth constrained minimization duality subgradient method stopping rule convergence.

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A SQP METHOD FOR MINIMIZING A CLASS OF nonsmooth FUNCTIONS

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Numerical Mathematics A Journal of Chinese Universities(English Series) 1996年第2期5卷 139-146页

作者：孙小玲张连生白延琴 Department of Mathematics Shanghai UniversityShanghai JiadingShanhai 201800PRC. Department of Mathematics Shanghai universityShanghai JiadingShanhai 201800PRC.

In this paper,we present a successive quadratic programming(SQP)method for minimizing a class of nonsmooth functions,which are the sum of a convex function and a nonsmooth composite function.The method generates new iterations by using the Armijo-type line search technique after having found the search directions.Global convergence property is established under mild assumptions.Numerical results are also offered.

关键词： nonsmooth optimization SQP method global convergence.

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