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The variable metric three-operator algorithms for solving monotone inclusions

OPTIMIZATION 2025年第3期74卷 751-779页

作者： Guo, Yanni Yan, Yinan Civil Aviat Univ China Coll Sci Tianjin Peoples R China

In this paper, we propose an inexact variable metric three-operator algorithm and a viscous version of it for finding a zero of the sum of three monotone operators in a real Hilbert space. Under some appropriate conditions, we establish the weak or strong convergence of the proposed algorithms. As a special case of the proposed problem, we present a variable metric Douglas-Rachford splitting algorithm and obtain a strong convergence result. The algorithms and the results in this paper are further extensions of the algorithms and results for solving the zero inclusion problems of monotone operators. To illustrate the efficiency of the proposed algorithms, we construct the numerical experiments and give some choices for the variable metric matrix.

关键词： Strong convergence variable metric inexact algorithm three-operator splitting monotone inclusions

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Two Efficient inexact algorithms for a Class of Large Sparse Complex Linear Systems

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MEDITERRANEAN JOURNAL OF MATHEMATICS 2016年第4期13卷 2301-2318页

作者： Edalatpour, Vahid Hezari, Davod Salkuyeh, Davod Khojasteh Univ Guilan Fac Math Sci Rasht Iran

Recently Salkuyeh et al. (Int J Comput Math 92:802-815, 2015) studied the generalized SOR (GSOR) iterative method for a class of complex symmetric linear system of equations. In this paper, we present an inexact variant of the GSOR method in which the conjugate gradient and the preconditioned conjugate gradient methods are regarded as its inner iteration processes at each step of the GSOR outer iteration. Moreover, we construct a new method called shifted GSOR iteration method which is obtained from combination of a shift-splitting iteration scheme and the GSOR iteration method. The convergence analysis of the proposed methods are presented. Some numerical experiments are given to show the performance of the methods and are compared with those of the inexact MHSS method.

关键词： Complex symmetric systems real equivalent form inexact algorithm shift splitting GSOR method

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A POLYNOMIAL-TIME INTERIOR-POINT METHOD FOR CONIC OPTIMIZATION, WITH inexact BARRIER EVALUATIONS

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SIAM JOURNAL ON OPTIMIZATION 2009年第1期20卷 548-571页

作者： Schurr, Simon P. O'Leary, Dianne P. Tits, Andre L. Univ Waterloo Dept Combinator & Optimizat Waterloo ON N2L 3G1 Canada Univ Maryland Dept Comp Sci College Pk MD 20742 USA Univ Maryland Inst Adv Comp Studies College Pk MD 20742 USA Univ Maryland Dept Elect & Comp Engn College Pk MD 20742 USA Univ Maryland Syst Res Inst College Pk MD 20742 USA

We consider a primal-dual short-step interior-point method for conic convex optimization problems for which exact evaluation of the gradient and Hessian of the primal and dual barrier functions is either impossible or prohibitively expensive. As our main contribution, we show that if approximate gradients and Hessians of the primal barrier function can be computed, and the relative errors in such quantities are not too large, then the method has polynomial worst-case iteration complexity. (In particular, polynomial iteration complexity ensues when the gradient and Hessian are evaluated exactly.) In addition, the algorithm requires no evaluation-or even approximate evaluation-of quantities related to the barrier function for the dual cone, even for problems in which the underlying cone is not self-dual.

关键词： convex cone conic optimization self-concordant barrier function interior-point method inexact algorithm polynomial complexity

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inexact and incremental bilinear Lanczos components algorithms for high dimensionality reduction and image reconstruction

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PATTERN RECOGNITION 2015年第1期48卷 244-263页

作者： Wu, Gang Xu, Wei Leng, Huan China Univ Min & Technol Dept Math Xuzhou 221116 Jiangsu Peoples R China Jiangsu Normal Univ Sch Math & Stat Xuzhou 221116 Jiangsu Peoples R China Tongji Univ Dept Math Shanghai 200092 Peoples R China Shuanggou Middle Sch Suining Country Suining 221212 Jiangsu Peoples R China

Recently, matrix-based methods have gained wide attentions in pattern recognition and machine learning communities. The generalized low rank approximations of matrices (GLRAM) and the bilinear Lanczos components (BLC) algorithm are two popular algorithms that treat data as the native two-dimensional matrix patterns. However, these two algorithms often require heavy computation time and memory space in practice, especially for large scale problems. In this paper, we propose inexact and incremental bilinear Lanczos components algorithms for high dimensionality reduction and image reconstruction. We first introduce the thick-restarting strategy to the BLC algorithm, and present a thick-restarted Lanczos components (TRBLC) algorithm. In this algorithm, we use the Ritz vectors as approximations to dominant eigenvectors instead of the Lanczos vectors. In our implementation, the iterative matrices are neither formed nor stored explicitly, thanks to the characteristic of the Lanczos procedure. Then, we explore the relationship between the reconstruction error and the accuracy of the Ritz vectors, so that the computational complexities of eigenpairs can be reduced significantly. As a result, we propose an inexact thick-restarted Lanczos components (Inex-TRBLC) algorithm. Moreover, we investigate the problem of incremental generalized low rank approximations of matrices, and propose an incremental and inexact TRBLC (Incr-TRBLC) algorithm. Numerical experiments illustrate the superiority of the new algorithms over the GLRAM algorithm and its variations, as well as the BLC algorithm for some real-world image reconstruction and face recognition problems. (C) 2014 Elsevier Ltd. All rights reserved.

关键词： Dimensionality reduction Image reconstruction Generalized Low Rank Approximations of Matrices (GLRAM) Bilinear Lanczos components (BLC) algorithm inexact algorithm Incremental learning problem

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Approximate proximal methods in vector optimization

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2007年第1期183卷 1-19页

作者： Ceng, Lu-Chuan Yao, Jen-Chih Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 80424 Taiwan Shanghai Normal Univ Dept Math Shanghai 200234 Peoples R China

This paper studies the vector optimization problem of finding weakly efficient points for mappings in a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C subset of Y with a nonempty interior. The proximal method in vector optimization is extended to develop an approximate proximal method for this problem by virtue of the approximate proximal point method for finding a root of a maximal monotone operator. In this approximate proximal method, the subproblems consist of finding weakly efficient points for suitable regularizations of the original mapping. We present both an absolute and a relative version, in which the subproblems are solved only approximately. Weak convergence of the generated sequence to a weak efficient point is established. In addition, we also discuss an extension to Bregman-function-based proximal algorithms for finding weakly efficient points for mappings. (c) 2006 Elsevier B.V. All rights reserved.

关键词： vector optimization proximal point inexact algorithm Banach space

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An inexactly accelerated algorithm for nonnegative tensor CP decomposition with the column unit constraints

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2024年第3期88卷 923-962页

作者： Wang, Zihao Bai, Minru Hunan Univ Sch Math Changsha 410082 Hunan Peoples R China

The component separation problem in complex chemical systems is very important and challenging in chemometrics. In this paper, we study a third-order nonnegative CANDECOMP/PARAFAC decomposition model with the column unit constraints (NCPD_CU) motivated by the component separation problem. To solve the NCPD_CU model, we first explore rapid computational methods for a generalized class of three-block optimization problems, which may exhibit nonconvexity and nonsmoothness. To this end, we propose the accelerated inexact block coordinate descent (AIBCD) algorithm, where each subproblem is inexactly solved through a finite number of inner-iterations employing the alternating proximal gradient method. Additionally, the algorithm incorporates extrapolation during the outer-iterations to enhance overall efficiency. We prove that the iterative sequence generated by the algorithm converges to a stationary point under mild conditions. Subsequently, we apply this methodology to the NCPD_CU model that satisfies the specified conditions. Finally, we present numerical results using both synthetic and real-world data, showcasing the remarkable efficiency of our proposed method.

关键词： Accelerated block coordinate descent inexact algorithm Nonnegative CANDECOMP/PARAFAC decomposition Column unit constraint Component separation

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Proximal methods in vector optimization

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SIAM JOURNAL ON OPTIMIZATION 2005年第4期15卷 953-970页

作者： Bonnel, H Iusem, AN Svaiter, BF Univ Nouvelle Caledonie ERIM Noumea 98847 New Caledonia Inst Matematica Pura & Aplicada BR-22460320 Rio De Janeiro Brazil

We consider the vector optimization problem of finding weakly efficient points for maps from a Hilbert space X to a Banach space Y, with respect to the partial order induced by a closed, convex, and pointed cone C subset of Y with a nonempty interior. We develop for this problem an extension of the proximal point method for scalar-valued convex optimization. In this extension, the subproblems consist of finding weakly efficient points for suitable regularizations of the original map. We present both an exact and an inexact version, in which the subproblems are solved only approximately, within a constant relative tolerance. In both cases, we prove weak convergence of the generated sequence to a weakly efficient point, assuming convexity of the map with respect to C and C-completeness of the initial section. In cases where this last assumption fails, we still establish that the generating sequence is, in a suitable sense, a minimizing one. We also exhibit a particular instance of the algorithm for which, under a mild hypothesis on C, the weak limit of the generated sequence is an efficient, rather than a weakly efficient, point.

关键词： vector optimization proximal point inexact algorithm

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Convergence Analysis of Dual Decomposition algorithm in Distributed Optimization: Asynchrony and inexactness

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2023年第8期68卷 4767-4782页

作者： Su, Yifan Wang, Zhaojian Cao, Ming Jia, Mengshuo Liu, Feng Tsinghua Univ Dept Elect Engn State Key Lab Power Syst Beijing 100084 Peoples R China Shanghai Jiao Tong Univ Dept Automat Key Lab Syst Control & Informat Proc Minist Educ China Shanghai 200240 Peoples R China Univ Groningen Fac Sci & Engn NL-9747 AG Groningen Netherlands Swiss Fed Inst Technol Power Syst Lab CH-8092 Zurich Switzerland

Dual decomposition is widely utilized in the distributed optimization of multiagent systems. In practice, the dual decomposition algorithm is desired to admit an asynchronous implementation due to imperfect communication, such as time delay and packet drop. In addition, computational errors also exist when the individual agents solve their own subproblems. In this article, we analyze the convergence of the dual decomposition algorithm in the distributed optimization when both the communication asynchrony and the subproblem solution inexactness exist. We find that the interaction between asynchrony and inexactness slows down the convergence rate from O(1/k) to O(1/v/k). Specifically, with a constant step size, the value of the objective function converges to a neighborhood of the optimal value, and the solution converges to a neighborhood of the optimal solution. Moreover, the violation of v/ the constraints diminishes in O(1/ k). Our result generalizes and unifies the existing ones that only consider either asynchrony or inexactness. Finally, numerical simulations validate the theoretical results.

关键词： Asynchronous algorithm distributed optimization dual decomposition inexact algorithm multiagent system (MAS)

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Hierarchically Distributed Energy Management in Distribution Systems: An Error-Tolerant and Asynchronous Approach

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IEEE TRANSACTIONS ON SMART GRID 2024年第3期15卷 2909-2920页

作者： Su, Yifan Wu, Shiyong Wang, Zhaojian Bao, Zhiyuan Liu, Feng Tsinghua Univ Dept Elect Engn State Key Lab Power Syst Operat & Control Beijing 100084 Peoples R China Shanghai Jiao Tong Univ Dept Automat Key Lab Syst Control & Informat Proc Minist Educ China Shanghai 200240 Peoples R China

Distribution systems are becoming more complex with the integration of numerous distributed energy resources (DERs), such as distributed generators (DGs), distributed energy storages (DESs), and controllable end-users (EUs). Aggregators (Aggs) are emerging to bridge the gap between the distribution system operator (DSO) and EUs. This creates a hierarchical "DSO-Agg-EU" architecture that requires effective and efficient coordination among massive agents. However, this architecture faces challenges such as communication asynchrony and solution inexactness among different agents, which may undermine the performance of energy management or even cause unexpected failures. To address these challenges, this paper proposes an error-tolerant and asynchronous algorithm based on dual decomposition for hierarchically distributed energy management (HDEM). The algorithm is theoretically guaranteed to converge to a near-optimal value and can be further accelerated by Nesterov's method. The algorithm is tested on the IEEE 123-bus distribution system and demonstrates high effectiveness and efficiency under asynchrony and inexactness.

关键词： Energy management hierarchical architecture distributed optimization dual decomposition asynchronous algorithm inexact algorithm

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Convergence rates of accelerated proximal gradient algorithms under independent noise

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NUMERICAL algorithmS 2019年第2期81卷 631-654页

作者： Sun, Tao Barrio, Roberto Jiang, Hao Cheng, Lizhi Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China Univ Zaragoza Dept Matemat Aplicada IUMA Grp Pedro Cerbuna 12 E-50009 Zaragoza Spain Univ Zaragoza Dept Matemat Aplicada CoDy Grp Pedro Cerbuna 12 E-50009 Zaragoza Spain Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China Natl Univ Def Technol Coll Sci Changsha 410073 Hunan Peoples R China Natl Univ Def Technol State Key Lab High Performance Computat Changsha 410073 Hunan Peoples R China

We consider an accelerated proximal gradient algorithm for the composite optimization with independent errors (errors little related with historical information) for solving linear inverse problems. We present a new inexact version of FISTA algorithm considering deterministic and stochastic noises. We prove some convergence rates of the algorithm and we connect it with the current existing catalyst framework for many algorithms in machine learning. We show that a catalyst can be regarded as a special case of the FISTA algorithm where the smooth part of the function vanishes. Our framework gives a more generic formulation that provides convergence results for the deterministic and stochastic noise cases and also to the catalyst framework. Some of our results provide simpler alternative analysis of some existing results in literature, but they also extend the results to more generic situations.

关键词： Proximal gradient algorithm inexact algorithm Independent noise Convergence rates Acceleration

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