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STRONGLY CONVERGENT INERTIAL forward-backward-forward algorithm WITHOUT ON-LINE RULE FOR VARIATIONAL INEQUALITIES

Acta Mathematica Scientia 2024年第2期44卷 551-566页

作者：姚永红 Abubakar ADAMU Yekini SHEHU School of Mathematical Sciences Tiangong UniversityTianjin300387China Center for Advanced Information Technology Kyung Hee UniversitySeoul02447South Korea Operational Research Center in Healthcare Near East UniversityTRNC Mersion 10Nicosia99138Turkey Charles Chidume Mathematics Institute African University of Science and TechnologyAbuja900107Nigeria School of Mathematical Sciences Zhejiang Normal UniversityJinhua321004China

This paper studies a strongly convergent inertial forward-backward-forward algorithm for the variational inequality problem in Hilbert *** our convergence analysis,we do not assume the on-line rule of the inertial parameters and the iterates,which have been assumed by several authors whenever a strongly convergent algorithm with an inertial extrapolation step is proposed for a variational inequality ***,our proof arguments are different from what is obtainable in the relevant ***,we give numerical tests to confirm the theoretical analysis and show that our proposed algorithm is superior to related ones in the literature.

关键词： forward-backward-forward algorithm inertial extrapolation variational inequality on-line rule

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A Relaxed Inertial forward-backward-forward algorithm for Solving Monotone Inclusions with Application to GANs

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JOURNAL OF MACHINE LEARNING RESEARCH 2023年第1期24卷 1-37页

作者： Bot, Radu I. Sedlmayer, Michael Vuong, Phan Tu Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Univ Vienna Res Network Data Sci Uni Vienna Kolingasse 14-16 A-1090 Vienna Austria Univ Southampton Math Sci Southampton SO17 1BJ England

We introduce a relaxed inertial forward-backward-forward (RIFBF) splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single valued monotone and Lipschitz continuous operator. This work aims to extend Tseng's forward-backward-forward method by both using inertial effects as well as relaxation parameters. We formulate first a second order dynamical system that approaches the solution set of the monotone inclusion problem to be solved and provide an asymptotic analysis for its trajectories. We provide for RIFBF, which follows by explicit time discretization, a convergence analysis in the general monotone case as well as when applied to the solving of pseudo-monotone variational inequalities. We illustrate the proposed method by applications to a bilinear saddle point problem, in the context of which we also emphasize the interplay between the inertial and the relaxation parameters, and to the training of Generative Adversarial Networks (GANs).

关键词： forward-backward-forward algorithm inertial effects relaxation parameters continuous time approach application to GANs

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A relaxed inertial forward-backward-forward algorithm for solving monotone inclusions with application to GANs

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2023年第1期24卷 191-227页

作者： Radu I. Boţ Michael Sedlmayer Phan Tu Vuong Faculty of Mathematics University of Vienna Vienna Austria Research Network Data Science @ Uni Vienna University of Vienna Vienna Austria Mathematical Sciences University of Southampton Southampton UK

We introduce a relaxed inertial forward-backward-forward (RIFBF) splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitz continuous operator. This work aims to extend Tseng's forward-backward-forward method by both using inertial effects as well as relaxation parameters. We formulate first a second order dynamical system that approaches the solution set of the monotone inclusion problem to be solved and provide an asymptotic analysis for its trajectories. We provide for RIFBF, which follows by explicit time discretization, a convergence analysis in the general monotone case as well as when applied to the solving of pseudo-monotone variational inequalities. We illustrate the proposed method by applications to a bilinear saddle point problem, in the context of which we also emphasize the interplay between the inertial and the relaxation parameters, and to the training of Generative Adversarial Networks (GANs).

关键词： forward-backward-forward algorithm inertial effects relaxation parameters continuous time approach application to GANs

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Almost sure convergence of the forward-backward-forward splitting algorithm

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OPTIMIZATION LETTERS 2016年第4期10卷 781-803页

作者： Vu, Bang Cong Ist Italiano Tecnol LCSL Genoa Italy MIT Bldg 46-515577 Massachusetts Ave Cambridge MA 02139 USA

In this paper, we propose a stochastic forward-backward-forward splitting algorithm and prove its almost sure weak convergence in real separable Hilbert spaces. Applications to composite monotone inclusion and minimiz... 详细信息

关键词： Monotone inclusion Monotone operator Operator splitting Lipschitzian operators forward-backward-forward algorithm Composite operator Duality Primal-dual algorithm

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forward-backward-forward algorithms involving two inertial terms for monotone inclusions

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COMPUTATIONAL & APPLIED MATHEMATICS 2023年第6期42卷 1-17页

作者： Suantai, Suthep Inkrong, Papatsara Cholamjiak, Prasit Chiang Mai Univ Fac Sci Res Ctr Optimizat & Computat Intelligence Big Data Dept Math Chiang Mai 50200 Thailand Univ Phayao Sch Sci Phayao 56000 Thailand

We come up with a new type of forward-backward-forward algorithms for monotone inclusion problems based on a self-adaptive technique to avoid the selection of Lipschitz assumption and also double inertial extrapolations to increase the convergence performance of our presented algorithm. We also prove its weak convergence theorem under mild hypothesis. Additionally, we provide numerical test in image deblurring and signal recovery as applications. The results show that our algorithm outperforms some known algorithms in the literature.

关键词： forward-backward-forward algorithm Inclusion problem Inertial method Weak convergence

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A distributed stochastic forward-backward-forward self-adaptive algorithm for Cartesian stochastic variational inequalities

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APPLIED NUMERICAL MATHEMATICS 2025年 211卷 17-41页

作者： Liu, Liya Qin, Xiaolong Yao, Jen-Chih Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China Hangzhou Normal Univ Dept Math Zhejiang 311121 Peoples R China China Med Univ Ctr Gen Educ Taichung 40402 Taiwan

In this paper, we consider a Cartesian stochastic variational inequality with a high dimensional solution space. This mathematical formulation captures a wide range of optimization problems including stochastic Nash games and stochastic minimization problems. By combining the advantages of the forward-backward-forward method and the stochastic approximated method, a novel distributed algorithm is developed for addressing this large-scale problem without any kind of monotonicity. A salient feature of the proposed algorithm is to compute two independent queries of a stochastic oracle at each iteration. The main contributions include: (i) The necessary condition imposed on the involved operator is related merely to the Lipschitz continuity, which are quite general. (ii) At each iteration, the suggested algorithm only requires one computation of the projection onto each feasible set, which can be easily evaluated. (iii) The distributed implementation of the stochastic approximation based Armijo-type line search strategy is adopted to weaken the line search condition and define variable adaptive non-monotonic stepsizes, when the Lipschitz constant is unknown. Some theoretical results of the almost sure convergence, the optimal rate statement, and the oracle complexity bound are established with conditions weaker than the conditions of other methods studied in the literature. Finally, preliminary numerical results are presented to show the efficiency and the competitiveness of our algorithm.

关键词： Cartesian stochastic variational inequality forward-backward-forward algorithm Stochastic approximation Convergence rate Line search strategy

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Modified proximal gradient methods involving double inertial extrapolations for monotone inclusion

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2024年第15期47卷 12132-12148页

作者： Inkrong, Papatsara Cholamjiak, Prasit Univ Phayao Sch Sci Phayao Thailand Univ Phayao Sch Sci Phayao 56000 Thailand

In this work, we propose a novel class of forward-backward-forward algorithms for solving monotone inclusion problems. Our approach incorporates a self-adaptive technique to eliminate the need for explicitly selecting Lipschitz assumptions and utilizes two-step inertial extrapolations to enhance the convergence rate of the algorithm. We establish a weak convergence theorem under mild assumptions. Furthermore, we conduct numerical tests on image deblurring and data classification as practical applications. The experimental results demonstrate that our algorithm surpasses some existing methods in the literature which shows its superior performance and effectiveness.

关键词： forward-backward-forward algorithm inclusion problem inertial extrapolation weak convergence

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Iterative approximation of common solution to variational inequality problems in Hadamard manifold

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APPLICABLE ANALYSIS 2024年第14期103卷 2572-2590页

作者： Oyewole, Olawale K. Abass, Hammed A. Shehu, Yekini Technion Israel Inst Technol Dept Math Haifa Israel Sefako Makgatho Hlth Sci Univ Dept Math & Appl Math Pretoria South Africa Zhejiang Normal Univ Sch Math Sci Jinhua Peoples R China

The aim of this paper is to introduce a forward-backward-forward algorithm with inertial extrapolation to solve the problem of finding a common solution to the variational inequality problem (CSVIP) in a Hadamard manifold. Using a self-adaptive step size, we obtain convergence results under some standard conditions. Numerical examples are given to illustrate the theoretical analysis. Our result is a generalization and extension of previously announced results in this direction in the literature.

关键词： Variational inequality forward-backward-forward algorithm inertial extrapolation Hadamard manifold

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Convergent Noisy forward-backward-forward algorithms in non-monotone variational inequalities 15th

Convergent Noisy forward-backward-forward algorithms in non-...

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15th International-Federation-of-Automatic-Control (IFAC) Symposium on Large Scale Complex Systems (IFAC LSS)

作者： Staudigl, Mathias Mertikopoulos, Panayotis Maastricht Univ Dept Quantitat Econ Maastricht Netherlands Univ Grenoble Alpes CNRS INRIA LIG Grenoble France

We develop a new stochastic algorithm with variance reduction for solving pseudo monotone stochastic variational inequalities. Our method builds on Tseng's forward-backward forward algorithm, which is known in the deterministic literature to be a valuable alternative to Korpelevich's extragradient method when solving variational inequalities over a convex and closed set governed with pseudo-monotone and Lipschitz continuous operators. The main computational advantage of Tseng's algorithm is that it relies only on a single projection step, and two independent queries of a stochastic oracle. Our algorithm incorporates a variance reduction mechanism, and leads to a.s. convergence to solutions of a merely pseudo-monotone stochastic variational inequality problem. To the best of our knowledge, this is the first stochastic algorithm achieving this by using only a single projection at each iteration. (C) 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

关键词： Variational inequalites forward-backward-forward algorithm Stochastic Approximation Variance Reduction

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Convergent Noisy forward-backward-forward algorithms in non-monotone variational inequalities ⁎

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IFAC-PapersOnLine 2019年第3期52卷 120-125页

作者： Mathias Staudigl Panayotis Mertikopoulos Department of Quantitative Economics Maastricht University Maastricht The Netherlands Univ. Grenoble Alpes CNRS INRIA LIG Grenoble France

We develop a new stochastic algorithm with variance reduction for solving pseudo-monotone stochastic variational inequalities. Our method builds on Tseng’s forward-backward-forward algorithm, which is known in the deterministic literature to be a valuable alternative to Korpelevich’s extragradient method when solving variational inequalities over a convex and closed set governed with pseudo-monotone and Lipschitz continuous operators. The main computational advantage of Tseng’s algorithm is that it relies only on a single projection step, and two independent queries of a stochastic oracle. Our algorithm incorporates a variance reduction mechanism, and leads to a.s. convergence to solutions of a merely pseudo-monotone stochastic variational inequality problem. To the best of our knowledge, this is the first stochastic algorithm achieving this by using only a single projection at each iteration.

关键词： Variational inequalites forward-backward-forward algorithm Stochastic Approximation Variance Reduction

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