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The modified proximal point algorithm in Hadamard spaces

JOURNAL OF INEQUALITIES AND APPLICATIONS 2018年第1期2018卷 124-124页

作者： Chang, Shih-sen Wang, Lin Wen, Ching-Feng Zhang, Jian Qiang China Med Univ Ctr Gen Educ Taichung Taiwan Yunnan Univ Finance & Econ Coll Stat & Math Kunming Yunnan Peoples R China Kaohsiung Med Univ Ctr Fundamental Sci Kaohsiung Taiwan

The purpose of this paper is to propose a modified proximal point algorithm for solving minimization problems in Hadamard spaces. We then prove that the sequence generated by the algorithm converges strongly (convergence in metric) to a minimizer of convex objective functions. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces.

关键词： Hadamard space CAT(0) space Moreau-Yosida resolvent Implicit iterative rule proximal point algorithm Variational inequality

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On the weak and strong convergence of the proximal point algorithm in reflexive Banach spaces

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OPTIMIZATION 2017年第9期66卷 1487-1494页

作者： Dadashi, Vahid Khatibzadeh, Hadi Islamic Azad Univ Sari Branch Dept Math Sari Iran Univ Zanjan Dept Math Zanjan Iran

In this paper, some proximal point algorithms ( PPAs) for maximal monotone operators in Banach spaces are considered. We obtain some results on the boundedness and the convergence of sequences generated by the PPAs with some assumptions. We can also get zero of maximal f-expansive operator and maximal f-monotone at zero operator using this method and then we apply it for finding a solution of the equilibrium problem.

关键词： proximal point algorithm maximal monotone operator phi-strongly monotone phi-generalized monotone phi-expansive operator equilibrium problems

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proximal point algorithm for a Common of Countable Families of Inverse Strongly Accretive Operators and Nonexpansive Mappings with Convergence Analysis

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MATHEMATICAL MODELLING AND ANALYSIS 2016年第1期21卷 95-118页

作者： Promluang, Khanittha Sitthithakerngkiet, Kanokwan Kumam, Poom KMUTT Dept Math Fac Sci 126 Pracha Uthit Rd Bangkok 10140 Thailand King Mongkuts Univ North Bangkok KMUTNB Fac Sci Appl Dept Math Nonlinear Dynam Anal Res Ctr 1518 Pracharat 1 Rd Bangkok 10800 Thailand KMUTT Theoret & Computat Sci TaCS Ctr Fac Sci Sci Lab Bldg126 Pracha Uthit Rd Bangkok 10140 Thailand China Med Univ 91 Hsueh Shih Rd Taichung 40402 Taiwan

In this paper, we investigate and analyze a proximal point algorithm via viscosity approximation method with error. This algorithm is introduced for finding a common zero point for a countable family of inverse strongly accretive operators and a countable family of nonexpansive mappings in Banach spaces. Our result can be extended to some well known results from a Hilbert space to a uniformly convex and 2 uniformly smooth Banach space. Finally, we establish the strong convergence theorems for the proximal point algorithm. Also, some illustrative numerical examples are presented.

关键词： proximal point algorithm Zero point inverse strongly accretive operator nonexpansive mapping

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proximal point algorithm for infinite pseudo-monotone bifunctions

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OPTIMIZATION 2016年第8期65卷 1629-1639页

作者： Khatibzadeh, Hadi Mohebbi, Vahid Univ Zanjan Dept Math Zanjan Iran

In this paper, first we study the weak convergence of the proximal point algorithm for an infinite family of equilibrium problems of pseudo-monotone type in Hilbert spaces. Then with additional conditions on the bifunctions, we prove the strong convergence for the family to a common equilibrium point. We also study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point of the family of infinite pseudo-monotone bifunctions without any additional assumptions on the bifunctions. A concrete example of a family of pseudo-monotone bifunctions is also presented.

关键词： Equilibrium problem pseudo-monotone bifunction proximal point algorithm weak convergence strong convergence Halpern's method

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A proximal point algorithm based on decomposition method for cone constrained multiobjective optimization problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2016年第1期65卷 289-308页

作者： Chen, Jiawei Ansari, Qamrul Hasan Liou, Yeong-Cheng Yao, Jen-Chih Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China Aligarh Muslim Univ Dept Math Aligarh Uttar Pradesh India King Fahd Univ Petr & Minerals Dept Math & Stat Dhahran Saudi Arabia Cheng Shiu Univ Dept Informat Management Kaohsiung 833 Taiwan China Med Univ Ctr Gen Educ Taichung 40402 Taiwan China Med Univ China Med Univ Hosp Res Ctr Interneural Comp Taichung 40402 Taiwan

By using auxiliary principle technique, a new proximal point algorithm based on decomposition method is suggested for computing a weakly efficient solution of the constrained multiobjective optimization problem (MOP) without assuming the nonemptiness of its solution set. The optimality conditions for (MOP) are derived by the Lagrangian function of its subproblem and corresponding mixed variational inequality. Some basic properties and convergence results of the proposed method are established under some mild assumptions. As an application, we employ the proposed method to solve a split feasibility problem. Finally, numerical results are also presented to illustrate the feasibility of the proposed algorithm.

关键词： Multiobjective optimization with cone constraints Mixed variational inequalities Split feasibility problems proximal point algorithm Auxiliary principle Decomposition method

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Error Sensitivity for Strongly Convergent Modifications of the proximal point algorithm

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2016年第3期168卷 901-916页

作者： Wang, Yamin Wang, Fenghui Xu, Hong-Kun Henan Normal Univ Sch Math & Informat Sci Xinxiang 453007 Peoples R China Henan Normal Univ Henan Engn Lab Big Data Stat Anal & Optimal Contr Sch Math & Informat Sci Xinxiang 453007 Peoples R China E China Univ Sci & Technol Dept Math Shanghai 200237 Peoples R China Luoyang Normal Univ Dept Math Luoyang 471022 Peoples R China Hangzhou Dianzi Univ Dept Math Hangzhou 310018 Zhejiang Peoples R China King Abdulaziz Univ Dept Math POB 80203 Jeddah 21589 Saudi Arabia

The proximal point algorithm plays an important role in finding zeros of maximal monotone operators. It has however only weak convergence in the infinite-dimensional setting. In the present paper, we provide two contraction-proximal point algorithms. The strong convergence of the two algorithms is proved under two different accuracy criteria on the errors. A new technique of argument is used, and this makes sure that our conditions, which are sufficient for the strong convergence of the algorithms, are weaker than those used by several other authors.

关键词： proximal point algorithm Contraction-proximal point algorithm Maximal monotone operator Resolvent Strong convergence

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New Augmented Lagrangian-Based proximal point algorithm for Convex Optimization with Equality Constraints

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2016年第1期171卷 251-261页

作者： Shen, Yuan Wang, Hongyong Nanjing Univ Finance & Econ Nanjing Jiangsu Peoples R China

The augmented Lagrangian method is a classic and efficient method for solving constrained optimization problems. However, its efficiency is still, to a large extent, dependent on how efficient the subproblem be solved. When an accurate solution to the subproblem is computationally expensive, it is more practical to relax the subproblem. Specifically, when the objective function has a certain favorable structure, the relaxed subproblem yields a closed-form solution that can be solved efficiently. However, the resulting algorithm usually suffers from a slower convergence rate than the augmented Lagrangian method. In this paper, based on the relaxed subproblem, we propose a new algorithm with a faster convergence rate. Numerical results using the proposed approach are reported for three specific applications. The output is compared with the corresponding results from state-of-the-art algorithms, and it is shown that the efficiency of the proposed method is superior to that of existing approaches.

关键词： Convex optimization proximal point algorithm Augmented Lagrangian Sparse optimization Compressed sensing

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A proximal point algorithm with Quasi-distance in Multi-objective Optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2016年第3期171卷 964-979页

作者： Rocha, Rogerio A. Oliveira, Paulo R. Gregorio, Ronaldo M. Souza, Michael Fed Univ Tocantins Tocantins Brazil Univ Fed Rio de Janeiro Rio De Janeiro Brazil Rural Fed Univ Rio de Janeiro Rio De Janeiro Brazil Univ Fed Ceara Ceara Brazil

In this paper, we present a generalized vector-valued proximal point algorithm for convex and unconstrained multi-objective optimization problems. Our main contribution is the introduction of quasi-distance mappings in the regularized subproblems, which has important applications in the computer theory and economics, among others. By considering a certain class of quasi-distances, that are Lipschitz continuous and coercive in any of their arguments, we show that any sequence generated by our algorithm is bounded and its accumulation points are weak Pareto solutions.

关键词： Multi-objective optimization proximal point algorithm Weak Pareto solution Quasi-distance

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A Self-adaptive proximal point algorithm for Signal Reconstruction in Compressive Sensing

A Self-adaptive Proximal Point Algorithm for Signal Reconstr...

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IEEE International Conference on Signal and Image Processing (ICSIP)

作者： Huai, Kaizhan Li, Yejun Ni, Mingfang Yu, Zhanke Wang, Xiaoguo PLA Univ Sci & Technol Inst Commun Engn Nanjing Jiangsu Peoples R China Xian Commun Inst Xian Shaanxi Peoples R China

ISBN: (纸本)9781509023776

Compressive sensing (CS) is a new framework for simulations sensing and compressive. How to reconstruct a sparse signal from limited measurements is the key problem in CS. For solving the reconstruction problem of a sparse signal, we proposed a self-adaptive proximal point algorithm (PPA). This algorithm can handle the sparse signal reconstruction by solving a substituted problem-l(1) problem. At last, the numerical results shows that the proposed method is more effective compared with the compressive sampling matching pursuit (CoSaMP).

关键词： compressive sensing signal reconstruction proximal point algorithm self-adaptive

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Convergence analysis of the proximal point algorithm for pseudo-monotone equilibrium problems

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OPTIMIZATION METHODS & SOFTWARE 2015年第6期30卷 1146-1163页

作者： Khatibzadeh, Hadi Mohebbi, Vahid Ranjbar, Sajad Univ Zanjan Dept Math Zanjan Iran

In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators.

关键词： equilibrium problems pseudo-monotone bifunctions proximal point algorithm weak convergence strong convergence Halpern method

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