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Adaptive sparse and dense hybrid representation with nonconvex optimization

Frontiers of Computer Science 2020年第4期14卷 65-78页

作者： Xuejun WANG Feilong CAO Wenjian WANG School of Computer Science and Technology Shanxi UniversityTaiyuan030006China School of Science China Jiliang UniversityHangzhou310018China Key Laboratory of Computational Intelligence and Chinese Information Processing of Ministry of Education Taiyuan030006China

Sparse representation has been widely used in signal processing,pattern recognition and computer vision *** achievements have been made in both theoretical researches and practical ***,there are two limitations on the application of *** is that sufficient training samples are required for each class,and the other is that samples should be *** order to alleviate above problems,a sparse and dense hybrid representation(SDR)framework has been proposed,where the training dictionary is decomposed into a class-specific dictionary and a non-class-specific *** putsℓ1 constraint on the coefficients of class-specific ***,it over-emphasizes the sparsity and overlooks the correlation information in class-specific dictionary,which may lead to poor classification *** overcome this disadvantage,an adaptive sparse and dense hybrid representation with non-convex optimization(ASDR-NO)is proposed in this *** trace norm is adopted in class-specific dictionary,which is different from general *** doing so,the dictionary structure becomes adaptive and the representation ability of the dictionary will be ***,a non-convex surrogate is used to approximate the rank function in dictionary decomposition in order to avoid a suboptimal solution of the original rank minimization,which can be solved by iteratively reweighted nuclear norm(IRNN)*** experiments conducted on benchmark data sets have verified the effectiveness and advancement of the proposed algorithm compared with the state-of-the-art sparse representation methods.

关键词： sparse representation trace norm nonconvex optimization low rank matrix recovery iteratively reweighted nuclear norm

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Convergence of Perry and Shanno's Memoryless Quasi-Newton Method for nonconvex optimization Problems

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运筹学学报 1997年第1期 22-28页

作者： JI-YE HAN GUANG-HUI LIU HONG-XIA YIN(Institution of Applied Mathematics, the Chinese Academy of Sciences, Beijing, 100080,CHINA)

In this paper, some sufficient conditions were given for the global convergenceo f Perry and Shanno’s memoryless quasi-Newton method with inexact inesearches fornonconvex optimization problems.

关键词： nonconvex optimization Perry and Shanno’s memoryless method convergence

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Convergence of ADMM for multi-block nonconvex separable optimization models

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Frontiers of Mathematics in China 2017年第5期12卷 1139-1162页

作者： Ke GUO Deren HAN David Z. W. WANG Tingting WU School of Mathematical Sciences and Key Laboratory for NSLSCS of Jiangsu Province Nanjing Normal University Nanjing 210023 China School of Civil and Environmental Engineering Nanyang Technological UniversitySingapore 639798 Singapore School of Science Nanjing University of Posts and Telecommunications Nanjing 210023China

For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers （ADMM） has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function~ ADMM or its direct extended version may not converge. In this paper, we consider the multi-block *** optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.

关键词： nonconvex optimization separable structure alternating directionmethod of rnultip!iers (.ADMM) Kurdyka-Lojasiewicz inequality

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A Bregman-style Partially Symmetric Alternating Direction Method of Multipliers for nonconvex Multi-block optimization

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Acta Mathematicae Applicatae Sinica 2023年第2期39卷 354-380页

作者： Peng-jie LIU Jin-bao JIAN Guo-dong MA School of Mathematics China University of Mining and TechnologyXuzhou 221116China College of Mathematics and Physics Center for Applied Mathematics of GuangxiGuangxi Minzu UniversityNanning 530006China

The alternating direction method of multipliers(ADMM)is one of the most successful and powerful methods for separable minimization *** on the idea of symmetric ADMM in two-block optimization,we add an updating formula for the Lagrange multiplier without restricting its position for multiblock ***,combining with the Bregman distance,in this work,a Bregman-style partially symmetric ADMM is presented for nonconvex multi-block optimization with linear constraints,and the Lagrange multiplier is updated twice with different relaxation factors in the iteration *** the suitable conditions,the global convergence,strong convergence and convergence rate of the presented method are analyzed and ***,some preliminary numerical results are reported to support the correctness of the theoretical assertions,and these show that the presented method is numerically effective.

关键词： nonconvex optimization multi-block optimization alternating direction method with multipliers Kurdyka-Lojasiewicz property convergence rate

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A Bregman-Style Improved ADMM and its Linearized Version in the nonconvex Setting:Convergence and Rate Analyses

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Journal of the Operations Research Society of China 2024年第2期12卷 298-340页

作者： Peng-Jie Liu Jin-Bao Jian Hu Shao Xiao-Quan Wang Jia-Wei Xu Xiao-Yu Wu School of Mathematics Jiangsu Center for Applied MathematicsChina University of Mining and TechnologyXuzhou 221116JiangsuChina School of Mathematics and Physics Center for Applied Mathematics of GuangxiGuangxi Minzu UniversityNanning 530006GuangxiChina Department of Civil and Environmental Engineering The Hong Kong Polytechnic UniversityHong KongChina School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105HunanChina

This work explores a family of two-block nonconvex optimization problems subject to linear *** first introduce a simple but universal Bregman-style improved alternating direction method of multipliers(ADMM)based on the iteration framework of ADMM and the Bregman ***,we utilize the smooth performance of one of the components to develop a linearized version of *** to the traditional ADMM,both proposed methods integrate a convex combination strategy into the multiplier update *** each proposed method,we demonstrate the convergence of the entire iteration sequence to a unique critical point of the augmented Lagrangian function utilizing the powerful Kurdyka–Łojasiewicz property,and we also derive convergence rates for both the sequence of merit function values and the iteration ***,some numerical results show that the proposed methods are effective and encouraging for the Lasso model.

关键词： nonconvex optimization Alternating direction method of multipliers Kurdyka-Lojasiewicz property Convergence rate

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PROXIMAL POINT ALGORITHM FOR MINIMIZATION OF DC FUNCTION

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Journal of Computational Mathematics 2003年第4期21卷 451-462页

作者： Wen-yuSun Raimundo.J.B.Sampaio M.A.B.Candido SchoolofMathematicsandComputerScience NanjingNormalUniversityNanjing2100g7China ProgramadePos-GraduacaoemInformaticaAplicada PontificiaUniversidadeCatolicadoParana(PUCPR)CEP:80215-g01CuritibaPRBrazil ProgramadePos-GraduacaoemInformaticaAplicada PontificiaUniversidadeCatolicadoParana(PUCPR)CEP:80215-g01CuritibaPRBrazil

In this paper we present some algorithms for minimization of DC function (difference of two convex functions). They are descent methods of the proximal-type which use the convex properties of the two convex functions separately. We also consider an approximate proximal point algorithm. Some properties of the ε-subdifferential and the ε-directional derivative are discussed. The convergence properties of the algorithms are established in both exact and approximate forms. Finally, we give some applications to the concave programming and maximum eigenvalue problems.

关键词： nonconvex optimization Nonsmooth optimization DC function Proximal point algorithm ε-subgradient.

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Online blind source separation based on joint diagonalization

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Journal of Systems Engineering and Electronics 2009年第2期20卷 229-233页

作者： Li Ronghua Zhou Guoxu Yang Zuyuan Xie Shengli School of Electronic & Information Engineering South China Univ. of Technology Guangzhou 510641 P. R. China

A new algorithm is proposed for joint diagonalization. With a modified objective function, the new algorithm not only excludes trivial and unbalanced solutions successfully, but is also easily optimized. In addition, with the new objective function, the proposed algorithm can work well in online blind source separation （BSS） for the first time, although this family of algorithms is always thought to be valid only in batch-mode BSS by far. Simulations show that it is a very competitive joint diagonalization algorithm.

关键词： blind source separation joint diagonalization nonconvex optimization

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SNIG PROPERTY OF MATRIX LOW-RANK FACTORIZATION MODEL

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Journal of Computational Mathematics 2018年第3期36卷 374-390页

作者： Hong Wang Xin Liu Xiaojun Chen Yaxiang Yuan Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

Recently, the matrix factorization model attracts increasing attentions in handling large-scale rank minimization problems, which is essentially a nonconvex minimization problem. Specifically, it is a quadratic least squares problem and consequently a quartic polynomial optimization problem. In this paper, we introduce a concept of the SNIG （＂Second-order Necessary optimality Implies Global optimality＂） condition which stands for the property that any second-order stationary point of the matrix factorization model must be a global minimizer. Some scenarios under which the SNIG condition holds are presented. Furthermore, we illustrate by an example when the SNIG condition may fail.

关键词： Low rank factorization nonconvex optimization Second-order optimality condition Global minimizer

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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

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Convergence of Gradient Algorithms for nonconvex C^(1+α) Cost Functions

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Chinese Annals of Mathematics,Series B 2023年第3期44卷 445-464页

作者： Zixuan WANG Shanjian TANG Department of Finance and Control Sciences Shanghai Center for Mathematical SciencesFudan UniversityShanghai200433China Department of Finance and Control Sciences School of Mathematical SciencesFudan UniversityShanghai200433China

This paper is concerned with convergence of stochastic gradient algorithms with momentum terms in the nonconvex setting.A class of stochastic momentum methods,including stochastic gradient descent,heavy ball and Nesterov’s accelerated gradient,is analyzed in a general framework under mild *** on the convergence result of expected gradients,the authors prove the almost sure convergence by a detailed discussion of the effects of momentum and the number of *** is worth noting that there are not additional restrictions imposed on the objective function and *** improvement over previous results is that the existing Lipschitz condition of the gradient is relaxed into the condition of H?lder *** a byproduct,the authors apply a localization procedure to extend the results to stochastic stepsizes.

关键词： Gradient descent methods nonconvex optimization Accelerated gradient descent Heavy-ball momentum

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