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Three-term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping

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APPLIED MATHEMATICS AND COMPUTATION 2011年第13期217卷 6315-6327页

作者： Iiduka, Hideaki Kyushu Inst Technol Network Design Res Ctr Chiyoda Ku Tokyo 1000011 Japan

Many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain nonexpansive mapping, and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set. This paper presents a novel gradient method with a three-term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems. It is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions. Numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm. (C) 2011 Elsevier Inc. All rights reserved.

关键词： convex optimization problem Fixed point set convex function Nonexpansive mapping Conjugate gradient method Three-term conjugate gradient method

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A USE OF CONJUGATE GRADIENT DIRECTION FOR THE convex optimization problem OVER THE FIXED POINT SET OF A NONEXPANSIVE MAPPING

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SIAM JOURNAL ON optimization 2009年第4期19卷 1881-1893页

作者： Iiduka, Hideaki Yamada, Isao Kyushu Inst Technol Network Design Res Ctr Chiyoda Ku Tokyo 1000011 Japan Tokyo Inst Technol Dept Commun & Integrated Syst Meguro Ku Tokyo 1528552 Japan

In this paper, we discuss the convex optimization problem over the fixed point set of a nonexpansive mapping. The main objective of the paper is to accelerate the hybrid steepest descent method for the problem. To this goal, we present a new iterative scheme that utilizes the conjugate gradient direction. Its convergence to the solution is guaranteed under certain assumptions. In order to demonstrate the effectiveness, performance, and convergence of our proposed algorithm, we present numerical comparisons of the algorithm with the existing algorithm.

关键词： convex optimization problem nonexpansive mapping fixed point hybrid steepest descent method conjugate gradient direction

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CONSTRAINED convex-optimization problemS - WELL-POSEDNESS AND STABILITY

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NUMERICAL FUNCTIONAL ANALYSIS AND optimization 1994年第7-8期15卷 889-907页

作者： KONSULOVA, AS REVALSKI, JP BULGARIAN ACAD SCI INST MATHBU-1113 SOFIABULGARIA

We study convex minimization problems in a Banach space with inequality constraints. Different notions of well-posedness (i.e. uniqueness of the solution and its stability) of such kind of problems are introduced and investigated, including relations between them. The notions and results are generalized for the case when the requirement for uniqueness of the solution is dropped.

关键词： convex FUNCTIONS convex optimization problem WELL-POSED optimization problem AW-TOPOLOGY STABILITY

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An Approximate Penalty Method with Descent for convex optimization problems

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RUSSIAN MATHEMATICS 2019年第7期63卷 41-55页

作者： Konnov, I. V. Kazan Fed Univ Dept Syst Anal & Informat Technol 18 Kremlyovskaya Str Kazan 420008 Russia

We propose a penalty method for general convex constrained optimization problems, where each auxiliary penalized problem is solved approximately with a special composite descent method. Direction finding choice in this method is found with the help of an equivalent equilibrium type problem. This allows one to keep the complete structure of the initial problem, although without nonlinear constraints and to simply calculate the descent direction in separable problems. Convergence of the method in primal and dual variables is established under rather weak assumptions.

关键词： convex optimization problem non-linear constraints penalty method descent method decomposition

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An extended projected residual algorithm for solving smooth convex optimization problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2022年 412卷

作者： La Cruz, William Univ Cent Venezuela Fac Ingn Dept Elect Comp & Control Caracas 1053 Venezuela

A projected residual algorithm for solving smooth convex optimization problems is presented. The proposed method is an extension of a residual algorithm for solving systems of nonlinear monotone equations introduced by La Cruz (2017), which uses in a systematic way the residual as a search direction combined with the Barzilai-Borwein's choice of the step size and a line search globalization strategy that does not impose the condition that the function value to decrease monotonically at every iteration. The global and R-sublinear convergence of the new method is established. With the aim of showing the advantages of the proposed global scheme an extensive set of numerical experiments including standard test problems and some specific applications are reported. (c) 2022 Elsevier B.V. All rights reserved.

关键词： convex optimization problem Spectral projected gradient method Nonmonotone line search Nonlinear monotone equations

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A Novel Decomposition Solution Approach for the Restoration problem in Distribution Networks

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IEEE TRANSACTIONS ON POWER SYSTEMS 2020年第5期35卷 3810-3824页

作者： Sekhavatmanesh, Hossein Cherkaoui, Rachid Ecole Polytech Fed Lausanne EPFL Power Syst Res Grp CH-1015 Lausanne Switzerland

The distribution network restoration problem is by nature a mixed integer and non-linear optimization problem due to the switching decisions and Optimal Power Flow (OPF) constraints, respectively. The link between these two parts involves logical implications modelled through big-M coefficients. The presence of these coefficients makes the relaxation of the mixed-integer problem using branch-and-bound method very poor in terms of computation burden. Moreover, this link inhibits the use of classical Benders algorithm in decomposing the problem because the resulting cuts will still depend on the big-M coefficients. In this paper, a novel decomposition approach is proposed for the restoration problem named Modified Combinatorial Benders (MCB). In this regard, the reconfiguration problem and the OPF problem are decomposed into master and sub problems, which are solved through successive iterations. In the case of a large outage area, the numerical results show that the MCB provides, within a short time (after a few iterations), a restoration solution with a quality that is close to the proven optimality when it can be exhibited.

关键词： optimization Circuit faults Load flow Switches Network topology Computational modeling convex optimization problem decomposition distribution network load pickup line switches reconfiguration restoration service

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Generalized robust linear discriminant analysis for jointly sparse learning

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APPLIED INTELLIGENCE 2024年第19期54卷 9508-9523页

作者： Zhu, Yufei Lai, Zhihui Gao, Can Kong, Heng Shenzhen Univ Coll Comp Sci & Software Engn Shenzhen 518060 Peoples R China Baoan Cent Hosp Shenzhen Dept Breast & Thyroid Surg Shenzhen 518102 Peoples R China

Linear discriminant analysis (LDA) is a well-known supervised method that can perform dimensionality reduction and feature extraction effectively. However, traditional LDA-based methods need to be turned into the trace ratio form to compute the closed-form solution, in which the within-class scatter matrix should be nonsingular. In this article, we design a new model named generalized robust linear discriminant analysis (GRLDA) method to tackle this disadvantage and improve the robustness. GRLDA uses L2,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}_{\mathrm{2,1}}$$\end{document}-norm on both loss functions to reduce the influence of outliers and on regularization term to obtain joint sparsity simultaneously. The intrinsic graph and the penalty graph are constructed to characterize the intraclass similarity and interclass separability, respectively. A novel optimization method is proposed to solve the proposed model, in which a quadratic problem on the Stiefel manifold is involved to avoid the inverse computation on a singular matrix. We also analyze the computational complexity rigorously. Finally, the experimental results on face, object, and medical images exhibit the superiority of GRLDA.

关键词： Generalized Robust Linear Discriminant Analysis (GRLDA) Feature extraction convex optimization problem

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A fast robust numerical continuation solver to a two-dimensional spectral estimation problem

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IET CONTROL THEORY AND APPLICATIONS 2022年第9期16卷 902-915页

作者： Zhu, Bin Liu, Jiahao Sun Yat Sen Univ Sch Intelligent Syst Engn Waihuan East Rd 132 Guangzhou 510006 Peoples R China

This paper presents a fast algorithm to solve a spectral estimation problem for two-dimensional random fields. The latter is formulated as a convex optimization problem with the Itakura-Saito pseudodistance as the objective function subject to the constraints of moment equations. The structure of the Hessian of the dual objective function is exploited in order to make possible a fast Newton solver. Then the Newton solver is incorporated to a predictor-corrector numerical continuation method which is able to produce a parametrized family of solutions to the moment equations. Two sets of numerical simulations are performed to test the algorithm and spectral estimator. The simulations on the frequency estimation problem show that our spectral estimator outperforms the classical windowed periodograms in the case of two hidden frequencies and has a higher resolution. The other set of simulations on system identification indicates that the numerical continuation method is more robust than Newton's method alone in ill-conditioned instances.

关键词： predictor-corrector methods fast Newton solver convex optimization problem spectral estimator moment equations convex programming 2D spectral estimation problem Newton's method numerical simulations 2D random fields Itakura-Saito pseudodistance Newton method spectral analysis dual objective function fast robust numerical continuation solver numerical analysis frequency estimation predictor-corrector numerical continuation method frequency estimation problem objective function subject

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On the stability domain estimation via a quadratic Lyapunov function: convexity and optimality properties for polynomial systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1996年第11期41卷 1650-1657页

作者： Tesi, A Villoresi, F Genesio, R Dipartimento di Sistemie Informatica Università di Firenze Florence Italy

The problem of estimating the stability domain of the origin of an n-order polynomial system is considered. Exploiting the structure of this class of systems it is shown that, for a given quadratic Lyapunov function, an estimate of the stability domain can be obtained by solving a suitable convex optimization problem. This estimate is shown to be optimal for an important subclass including both quadratic and cubic systems, and its accuracy in the general polynomial case is discussed via several examples.

关键词： Lyapunov methods convex programming nonlinear programming polynomials stability stability criteria convex optimization problem cubic systems polynomial systems quadratic Lyapunov function quadratic systems stability domain estimation Lyapunov methods convex programming quadratic system convex function stability region cubic system Liapunov function Polynomial Quadratic Nonlinear Programming Stability criteria

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Periodical sparse low-rank matrix estimation algorithm for fault detection of rolling bearings

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ISA TRANSACTIONS 2020年 101卷 366-378页

作者： Wang, Baoxiang Liao, Yuhe Ding, Chuancang Zhang, Xining Xi An Jiao Tong Univ Sch Mech Engn Shaanxi Key Lab Mech Prod Qual Assurance & Diagno Xian 710049 Shaanxi Peoples R China Xi An Jiao Tong Univ Educ Minist Modern Design & Rotor Bearing Syst Key Lab Xian 710049 Peoples R China Xi An Jiao Tong Univ Sch Mech Engn State Key Lab Mfg Syst Engn Xian 710049 Shaanxi Peoples R China

Early bearing fault detection is crucial to avoid catastrophic accidents. However, the repetitive defect impulses indicating bearing fault are buried in heavy background noise. In the paper, a novel periodical sparse low-rank (PSLR) matrix estimation algorithm is proposed for extracting repetitive transients from noisy signal. Concretely, periodical group sparsity and low-lank property of fault transients in time-frequency domain are first revealed, and then an optimization problem is proposed for simultaneously promoting these two properties. Meanwhile, to further highlight the sparsity of fault features, the non-convex penalty functions are incorporated into the optimization problem. Then, for solving the proposed optimization problem, an iterative algorithm is derived based on alternating direction method of multipliers (ADMM) and majorization-minimization (MM), in which the traditional soft-thresholding operation is replaced by the proposed Gini-guided fault information thresholding (FIT) scheme to enhance fault transient extraction. Finally, simulated and real signals confirm the performance of proposed PSLR in extracting defect impulses from noisy vibration signal. (c) 2020 ISA. Published by Elsevier Ltd. All rights reserved.

关键词： Periodical sparse low-rank (PSLR) matrix estimation convex optimization problem Gini-guided fault information thresholding (FIT) scheme Fault detection Rolling bearings

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