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Generalized forward-backward splitting with penalization for monotone inclusion problems

JOURNAL OF GLOBAL OPTIMIZATION 2019年第4期73卷 825-847页

作者： Nimana, Nimit Petrot, Narin Khon Kaen Univ Dept Math Fac Sci Khon Kaen 40002 Thailand Naresuan Univ Dept Math Fac Sci Phitsanulok 65000 Thailand Naresuan Univ Fac Sci Ctr Excellence Nonlinear Anal & Optimizat Phitsanulok 65000 Thailand

We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximally monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided that the condition corresponding to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a large-scale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems.

关键词： Monotone operator Monotone inclusion Hierarchical optimization forward-backward algorithm 47H05 65K05 65K10 90C25

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Strong convergence result of forward-backward splitting methods for accretive operators in banach spaces with applications

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2018年第1期112卷 71-87页

作者： Shehu, Yekini Cai, Gang Univ Nigeria Dept Math Nsukka Nigeria Chongqing Normal Univ Coll Math Sci Chongqing 401331 Peoples R China

Our interest in this paper is to prove strong convergence results for finding zeros of the sum of two accretive operators by utilizing a viscosity type forward-backward splitting method. We also discuss applications of this method to approximation of solution to certain integro-differential equation with generalized p-Laplacian operator. Our results complement many recent and important results in the literature.

关键词： Accretive operator Resolvent Zero point Splitting method forward-backward algorithm Banach space

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The Dynamic Programming Method of Stochastic Differential Game for Functional forward-backward Stochastic System

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MATHEMATICAL PROBLEMS IN ENGINEERING 2013年第1期2013卷 1-14页

作者： Ji, Shaolin Sun, Chuanfeng Wei, Qingmeng Shandong Univ Inst Financial Studies Jinan 250100 Peoples R China Shandong Univ Inst Math Jinan 250100 Peoples R China

This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differential equation (FBSDE). For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs). Applying the Girsanov transformation method introduced by Buckdahn and Li (2008), the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP), we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.

关键词： DYNAMIC programming STOCHASTIC differential equations DIFFERENTIAL games forward-backward algorithm STOCHASTIC systems HAMILTON-Jacobi-Bellman equation TRANSFORMATIONS (Mathematics)

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forward-backward Approach for 3D Event Localization Using Commodity Smartphones for Ubiquitous Context-Aware Applications in Civil and Infrastructure Engineering

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COMPUTER-AIDED CIVIL AND INFRASTRUCTURE ENGINEERING 2016年第4期31卷 245-260页

作者： Yoon, Hyungchul Ham, Youngjib Golparvar-Fard, Mani Spencer, Billie F., Jr. Univ Illinois Dept Civil & Environm Engn Urbana IL 61801 USA Florida Int Univ OHL Sch Construct Miami FL 33174 USA

An inexpensive and robust 3D localization system for tracking the position of a user in GPS- or WLAN-denied environments offers significant potential for improving decision-making tasks for civil and infrastructure engineering applications. To this end, an infrastructure-free approach for 3D event localization on commodity smartphones is presented. In the proposed method, the position of the user is continuously tracked based on the smartphone sensory data (the forward approach) until the user reaches a certain event. Here, an event location refers to the 3D location of a user conducting value-added activities such as tasks involved in emergency response and field reporting of operational issues. Once an event is observed, the motion trajectory of the user is backtracked from the postevent landmark to reestimate the location of the event (the backward approach). By integrating probability distributions of the forward and backward approaches together, the proposed method derives the most-likely location of the event. To validate the proposed approach, seven case studies are conducted in a multistory parking garage. The experimental results show that the probabilistic integration of the localization results from the forward and backward dead reckonings can produce more accurate 3D localization results when compared to a single best estimate from a one-way dead reckoning process. Lessons learned from several real-world case studies and open research challenges in improving localization accuracy are discussed in detail.

关键词： forward-backward algorithm DECISION making UBIQUITOUS computing DEAD reckoning (Navigation) WIRELESS LANs CIVIL engineering WEBSITES

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forward-backward study of the stereo vision problem

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COMPUTER PHYSICS COMMUNICATIONS 2009年第4期180卷 664-668页

作者： de Menezes, Fortunato Silva Geiger, Davi Abreu e Silva, Ricardo Martins Distler, George Hu, Arefin Univ Fed Lavras UFLA Dept Ciencias Exatas DEX Lavras MG Brazil NYU Courant Inst Math Sci Dept Comp Sci New York NY 10003 USA Univ Fed Lavras UFLA Dept Ciencias Comp DCC Lavras MG Brazil

In stereo vision a pair of two-dimensional (2D) stereo images is given and the purpose is to find out the depth (disparity) of regions of the image in relation to the background, so that we can reconstruct the 3D structure of the image from the pair of 2D stereo images given. Using the Bayesian framework we implemented the forward-backward algorithm to unfold the disparity (depth) of a pair of stereo images. The results showed are very reasonable, but we point out there is room for improvement concerning the graph structure used. (c) 2009 Elsevier B.V. All rights reserved.

关键词： Hidden Markov models forward-backward algorithm

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Range-Doppler Imaging via forward-backward Sparse Bayesian Learning

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2010年第4期58卷 2421-2425页

作者： Tan, Xing Li, Jian Univ Florida Dept Elect & Comp Engn Gainesville FL 32611 USA

We consider range-Doppler imaging via transmitting a train of probing pulses as in radar and active sonar. We show that range-Doppler imaging can be formulated as a sparse signal recovery problem and that we can use an expectation maximization based sparse Bayesian learning (EM-SBL) algorithm to achieve high resolution imaging. We also reduce the complexity of EM-SBL significantly by using an efficient forward-backward algorithm in the E step of the EM algorithm.

关键词： forward-backward algorithm radar imaging Range-Doppler imaging sparse Bayesian learning super resolution

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Shrinking projection methods involving inertial forward-backward splitting methods for inclusion problems

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REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2019年第2期113卷 645-656页

作者： Khan, Suhel Ahmad Suantai, Suthep Cholamjiak, Watcharaporn BITS Pilani Dept Math Dubai CampusPOB 345055 Dubai U Arab Emirates Chiang Mai Univ Fac Sci Dept Math Ctr Excellence Math & Appl Math Chiang Mai 50200 Thailand Univ Phayao Sch Sci Phayao 56000 Thailand

In this paper, we propose a modified forward-backward splitting method using the shrinking projection and the inertial technique for solving the inclusion problem of the sum of two monotone operators. We prove its strong convergence under some suitable conditions in Hilbert spaces. We provide some numerical experiments including a comparison to show the implementation and the efficiency of our method.

关键词： Shrinking projection method Inertial method Inclusion problem Maximal monotone operator forward-backward algorithm 47H04 47H10

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A modified forward-backward splitting methods for the sum of two monotone operators with applications to breast cancer prediction

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第1期46卷 1251-1265页

作者： Peeyada, Pronpat Dutta, Hemen Shiangjen, Kanokwatt Cholamjiak, Watcharaporn Univ Phayao Sch Sci Phayao 56000 Thailand Gauhati Univ Dept Math Gauhati India Univ Phayao Sch Informat & Commun Technol Phayao Thailand

This work proposes a modified forward-backward splitting algorithm combining an inertial technique for solving the monotone variational inclusion problem. The weak convergence theorem is established under some suitable conditions in Hilbert space, and a new step size is presented for our algorithm to speed up the convergence. We give an example and numerical results for supporting our main theorem in infinite dimensional spaces. We also provide an application to predict breast cancer by using our proposed algorithm for updating the optimal weight in machine learning. Moreover, we use the Wisconsin original breast cancer data set as a training set to show efficiency comparing with the other three algorithms in terms of three key parameters, namely, accuracy, recall, and precision.

关键词： breast cancer data classification forward-backward algorithm inertial method machine learning monotone inclusion

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Distributed forward-backward methods for ring networks

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第3期86卷 845-870页

作者： Aragon-Artacho, Francisco J. Malitsky, Yura Tam, Matthew K. Torregrosa-Belen, David Univ Alicante Dept Math Alicante 03690 Spain Linkoping Univ Dept Math S-58183 Linkoping Sweden Univ Melbourne Sch Math & Stat Parkville Vic 3010 Australia

In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in the product space. Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.

关键词： Monotone operator Monotone inclusion Splitting algorithm forward-backward algorithm Distributed optimisation

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A Reflected forward-backward Splitting Method for Monotone Inclusions Involving Lipschitzian Operators

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SET-VALUED AND VARIATIONAL ANALYSIS 2021年第1期29卷 163-174页

作者： Cevher, Volkan Bang Cong Vu Ecole Polytech Fed Lausanne Lab Informat & Inference Syst Lausanne Switzerland

In this paper, we propose a novel splitting method for finding a zero point of the sum of two monotone operators where one of them is Lipschizian. The weak convergence the method is proved in real Hilbert spaces. Applying the proposed method to composite monotone inclusions involving parallel sums yields a new primal-dual splitting which is different from the existing methods. Connections to existing works are clearly stated. We also provide an application of the proposed method to the image denoising by the total variation.

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

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