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JOURNAL OF NONLINEAR FUNCTIONAL ANALYSIS 2024年 2024卷

作者： Yao, Bingchen Wang, Yamin Shaoxing Univ Dept Math Shaoxing 312000 Peoples R China Henan Normal Univ Coll Math & Informat Sci Xinxiang 453007 Peoples R China

In the paper, we study a viscosity alternating resolvent algorithm with over relaxed factors for finding a common zero point of two maximal monotone operators. We give the strong convergence of the algorithm under some mild conditions and more error criteria. This algorithm can be seen as a of the results in some current literature.

关键词： Alternating resolvent algorithm Monotone operators Over relaxed factor proximal point algorithm Viscosity

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A New Stopping Criterion for Eckstein and Bertsekas’s Generalized Alternating Direction Method of Multipliers

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Journal of the Operations Research Society of China 2023年第4期11卷 941-955页

作者： Xin-Xin Li Xiao-Ya Zhang School of Mathematics Jilin UniversityChangchun130012JilinChina

In this paper,we propose a new stopping criterion for Eckstein and Bertsekas’s generalized alternating direction method of *** stopping criterion is easy to verify,and the computational cost is much less than the cla... 详细信息

关键词： Convex optimization Generalized alternating direction method of multipliers proximal point algorithm Stopping criterion

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A customized proximal point algorithm for convex minimization with linear constraints

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2013年第3期56卷 559-572页

作者： He, Bingsheng Yuan, Xiaoming Zhang, Wenxing Nanjing Univ Int Ctr Management Sci & Engn Nanjing 210093 Jiangsu Peoples R China Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Hong Kong Baptist Univ Dept Math Hong Kong Hong Kong Peoples R China Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Peoples R China

This paper demonstrates a customized application of the classical proximal point algorithm (PPA) to the convex minimization problem with linear constraints. We show that if the proximal parameter in metric form is chosen appropriately, the application of PPA could be effective to exploit the simplicity of the objective function. The resulting subproblems could be easier than those of the augmented Lagrangian method (ALM), a benchmark method for the model under our consideration. The efficiency of the customized application of PPA is demonstrated by some image processing problems.

关键词： Convex minimization proximal point algorithm Resolvent operator Augmented Lagrangian method

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A proximal point algorithm FOR LOG-DETERMINANT OPTIMIZATION WITH GROUP LASSO REGULARIZATION

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SIAM JOURNAL ON OPTIMIZATION 2013年第2期23卷 857-893页

作者： Yang, Junfeng Sun, Defeng Toh, Kim-Chuan Nanjing Univ Dept Math Nanjing 210093 Jiangsu Peoples R China Natl Univ Singapore Dept Math Singapore 119076 Singapore Natl Univ Singapore Risk Management Inst Singapore 119076 Singapore

We consider the covariance selection problem where variables are clustered into groups and the inverse covariance matrix is expected to have a blockwise sparse structure. This problem is realized via penalizing the maximum likelihood estimation of the inverse covariance matrix by group Lasso regularization. We propose to solve the resulting log-determinant optimization problem with the classical proximal point algorithm (PPA). At each iteration, as it is difficult to update the primal variables directly, we first solve the dual subproblem by an inexact semismooth Newton-CG method and then update the primal variables by explicit formulas based on the computed dual variables. We also propose to accelerate the PPA by an inexact generalized Newton's method when the iterate is close to the solution. Theoretically, we prove that at the optimal solution, the nonsingularity of the generalized Hessian matrices of the dual subproblem is equivalent to the constraint nondegeneracy condition for the primal problem. Global and local convergence results are also presented for the proposed PPA. Moreover, based on the augmented Lagrangian function of the dual problem we derive an alternating direction method (ADM), which is easily implementable and is demonstrated to be efficient for random problems. Numerical results, including comparisons with the ADM on both synthetic and real data, are presented to demonstrate that the proposed Newton-CG based PPA is stable and efficient and, in particular, outperforms the ADM when high accuracy is required.

关键词： covariance selection log-determinant optimization group Lasso regularization proximal point algorithm augmented Lagrangian alternating direction method Newton's method Gaussian graphical model

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A proximal point algorithm FOR SEQUENTIAL FEATURE EXTRACTION APPLICATIONS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2013年第1期35卷 A517-A540页

作者： Doan, Xuan Vinh Toh, Kim-Chuan Vavasis, Stephen Univ Warwick Warwick Business Sch DIMAP Coventry CV4 7AL W Midlands England Univ Warwick Warwick Business Sch ORMS Grp Coventry CV4 7AL W Midlands England Univ Waterloo Dept Combinator & Optimizat Waterloo ON N2L 3G1 Canada Natl Univ Singapore Dept Math Singapore 119076 Singapore

We propose a proximal point algorithm to solve the LAROS problem, that is, the problem of finding a "large approximately rank-one submatrix." This LAROS problem is used to sequentially extract features in data. We also develop new stopping criteria for the proximal point algorithm, which is based on the duality conditions of epsilon-optimal solutions of the LAROS problem, with a theoretical guarantee. We test our algorithm with two image databases and show that we can use the LAROS problem to extract appropriate common features from these images.

关键词： proximal point algorithm nuclear norm feature extraction

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Strong convergence of a proximal point algorithm with bounded error sequence

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OPTIMIZATION LETTERS 2013年第2期7卷 415-420页

作者： Boikanyo, Oganeditse A. Morosanu, Gheorghe Univ Botswana Dept Math Gaborone Botswana Cent European Univ Dept Math & Its Applicat H-1051 Budapest Hungary

Given any maximal monotone operator in a real Hilbert space H with , it is shown that the sequence of proximal iterates converges strongly to the metric projection of u on A (-1)(0) for (e (n) ) bounded, with and gamma (n) > 0 with as . In comparison with our previous paper (Boikanyo and MoroAYanu in Optim Lett 4(4):635-641, 2010), where the error sequence was supposed to converge to zero, here we consider the classical condition that errors be bounded. In the case when A is the subdifferential of a proper convex lower semicontinuous function , the algorithm can be used to approximate the minimizer of phi which is nearest to u.

关键词： Maximal monotone operator Nonexpansive map proximal point algorithm Prox-Tikhonov method Resolvent operator

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Strong Convergence of Regularized New proximal point algorithms

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第3期181卷 864-882页

作者： Rouhani, Behzad Djafari Moradi, Sirous Univ Texas El Paso Dept Math Sci El Paso TX 79968 USA Arak Univ Fac Sci Dept Math Arak *** Iran

We consider the regularization of two proximal point algorithms (PPA) with errors for a maximal monotone operator in a real Hilbert space, previously studied, respectively, by Xu, and by Boikanyo and Morosanu, where they assumed the zero set of the operator to be nonempty. We provide a counterexample showing an error in Xu's theorem, and then we prove its correct extended version by giving a necessary and sufficient condition for the zero set of the operator to be nonempty and showing the strong convergence of the regularized scheme to a zero of the operator. This will give a first affirmative answer to the open question raised by Boikanyo and Morosanu concerning the design of a PPA, where the error sequence tends to zero and a parameter sequence remains bounded. Then, we investigate the second PPA with various new conditions on the parameter sequences and prove similar theorems as above, providing also a second affirmative answer to the open question of Boikanyo and Morosanu. Finally, we present some applications of our new convergence results to optimization and variational inequalities.

关键词： Maximal monotone operator proximal point algorithm Resolvent operator Metric projection Hilbert space 47J25 47H05 47H09 90C29 90C90

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A proximal-proximal Majorization-Minimization algorithm for Nonconvex Rank Regression Problems

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2023年 71卷 3502-3517页

作者： Tang, Peipei Wang, Chengjing Jiang, Bo Hangzhou City Univ Sch Comp & Comp Sci Hangzhou 310015 Peoples R China Southwest Jiaotong Univ Sch Math Chengdu 611756 Peoples R China Zhejiang Univ Sch Comp Sci Hangzhou 310027 Zhejiang Peoples R China

In this paper, we introduce a proximal-proximal majorization-minimization (PPMM) algorithm for nonconvex rank regression problems. The basic idea of the algorithm is to apply the proximal majorization-minimization algorithm to solve the nonconvex problem with the inner subproblems solved by a sparse semismooth Newton (SSN) method based proximal point algorithm (PPA). It deserves mentioning that we adopt the sequential regularization technique and design an implementable stopping criterion to overcome the singular difficulty of the inner subproblem. Especially for the stopping criterion, it plays a very important role for the success of the algorithm. Furthermore, we also prove that the PPMM algorithm converges to a stationary point. Due to the Kurdyka-Lojasiewicz (KL) property of the problem, we present the convergence rate of the PPMM algorithm. Numerical experiments demonstrate that our proposed algorithm outperforms the existing state-of-the-art algorithms.

关键词： Nonconvex rank regression problems semismooth Newton method proximal-proximal majorization-minimization algorithm proximal point algorithm

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An alternative extrapolation scheme of PDHGM for saddle point problem with nonlinear function

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第1期85卷 263-291页

作者： Gao, Ying Zhang, Wenxing Univ Elect Sci & Technol China Sch Math Sci Chengdu Peoples R China

Primal-dual hybrid gradient (PDHG) method is a canonical and popular prototype for solving saddle point problem (SPP). However, the nonlinear coupling term in SPP excludes the application of PDHG on far-reaching real-world problems. In this paper, following the seminal work by Valkonen (Inverse Problems 30, 2014), we devise a variant iterative scheme for solving SPP with nonlinear function by exerting an alternative extrapolation procedure. The novel iterative scheme falls exactly into the proximal point algorithmic framework without any residuals, which indicates that the associated inclusion problem is "nearer" to the KKT mapping induced by SPP. Under the metrically regular assumption on KKT mapping, we simplify the local convergence of the proposed method on contractive perspective. Numerical simulations on a PDE-constrained nonlinear inverse problem demonstrate the compelling performance of the proposed method.

关键词： Primal-dual hybrid gradient (modified) proximal point algorithm Monotone inclusion Extrapolation Linearization Metric regularity Contraction

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proximal algorithm with quasidistances for multiobjective quasiconvex minimization in Riemannian manifolds

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RAIRO-OPERATIONS RESEARCH 2023年第4期57卷 2301-2314页

作者： Quiroz, Erik Alex Papa Rocha, Rogerio Azevedo Oliveira, Paulo Gregorio, Ronaldo Univ Privada Norte Univ Nacl Mayor San Marcos Lima Peru Univ Fed Goias Goiania Brazil Univ Fed Tocantins Palmas Brazil Univ Fed Rio de Janeiro Rio de Janeiro Brazil Univ Fed Rural Rio de Janeiro Rio De Janeiro Brazil

We introduce a proximal algorithm using quasidistances for multiobjective minimization problems with quasiconvex functions defined in arbitrary Riemannian manifolds. The reason of using quasidistances instead of the classical Riemannian distance comes from the applications in economy, computer science and behavioral sciences, where the quasidistances represent a non symmetric measure. Under some appropriate assumptions on the problem and using tools of Riemannian geometry we prove that accumulation points of the sequence generated by the algorithm satisfy the critical condition of Pareto-Clarke. If the functions are convex then these points are Pareto efficient solutions.

关键词： proximal point algorithm multiobjective minimization quasiconvex functions Riemannian manifolds quasidistances Pareto-Clarke critical point

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