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A First-Order Optimal Zero-Forcing Beamformer Design for Multiuser MIMO Systems via a Regularized Dual accelerated gradient method

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IEEE COMMUNICATIONS LETTERS 2015年第2期19卷 195-198页

作者： Li, Bin Dam, Hai Huyen Cantoni, Antonio Teo, Kok Lay Curtin Univ Dept Math & Stat Perth WA 6845 Australia Univ Western Australia Sch Elect Elect & Comp Engn Crawley WA 6009 Australia

A first-order zero-forcing beamformer design is proposed in this letter for MU-MIMO systems under per-antenna power constraints (PAPC). By forming the regularized dual problem, first-order methods can be applied. To achieve the limit convergence rate of first-order methods, an accelerated method is introduced to solve the regularized dual problem. Simulation results are provided to show the effectiveness of the proposed method.

关键词： Zero-forcing beamforming (ZFBF) per-antenna power constraint (PAPC) MIMO systems first-order method accelerated gradient method

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A timestamp-based Nesterov's accelerated projected gradient method for distributed Nash equilibrium seeking in monotone games

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SYSTEMS & CONTROL LETTERS 2024年 194卷

作者： Liu, Nian Tan, Shaolin Tao, Ye Lu, Jinhu Zhongguancun Lab Beijing 100094 Peoples R China Beihang Univ Sch Automat Sci & Elect Engn Beijing 100191 Peoples R China

In this paper, a timestamp-based Nesterov's accelerated gradient algorithm is proposed for Nash equilibrium seeking over communication networks for strongly monotone games. Its difference from the well-known consensus-based Nash equilibrium seeking method is that each player's local estimates of players' actions is updated by both Nesterov's accelerated gradient method and timestamp-based broadcasting protocol. We prove its convergence to the epsilon-approximation Nash equilibrium with the fixed step-size. Simulation results are given to demonstrate the outperformance of the proposed algorithm over some well-known projected gradient approaches. It is shown that the required number of iterations to reach the Nash equilibrium is greatly reduced in our proposed algorithm.

关键词： accelerated gradient method Timestamp-based projected play Nash equilibrium seeking Monotone game

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A feasible smoothing accelerated projected gradient method for nonsmooth convex optimization

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OPERATIONS RESEARCH LETTERS 2024年 57卷

作者： Nishioka, Akatsuki Kanno, Yoshihiro Univ Tokyo Dept Math Informat Bunkyo Ku Hongo 7-3-1Bunkyo Ku Tokyo 1138656 Japan Univ Tokyo Math & Informat Ctr Hongo 7-3-1Bunkyo Ku Tokyo 1138656 Japan

Smoothing accelerated gradient methods achieve faster convergence rates than that of the subgradient method for some nonsmooth convex optimization problems. However, Nesterov's extrapolation may require gradients at infeasible points, and thus they cannot be applied to some structural optimization problems. We introduce a variant of smoothing accelerated projected gradient methods where every variable is feasible. The O ( k - 1 log k ) convergence rate is obtained using the Lyapunov function. We conduct a numerical experiment on the robust compliance optimization of a truss structure. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

关键词： Smoothing method accelerated gradient method Convergence rate Structural optimization Eigenvalue optimization

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An inexact ADMM for separable nonconvex and nonsmooth optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2025年第2期90卷 445-479页

作者： Bai, Jianchao Zhang, Miao Zhang, Hongchao Northwestern Polytech Univ Shenzhen Res & Dev Inst Shenzhen 518057 Peoples R China Northwestern Polytech Univ Sch Math & Stat Xian 710072 Peoples R China Louisiana State Univ Dept Math Baton Rouge LA 70803 USA

An inexact alternating direction method of multiplies (I-ADMM) with an expansion linesearch step was developed for solving a family of separable minimization problems subject to linear constraints, where the objective function is the sum of a smooth but possibly nonconvex function and a possibly nonsmooth nonconvex function. Global convergence and linear convergence rate of the I-ADMM were established under proper conditions while inexact relative error criterion was used for solving the subproblems. In addition, a unified proximal gradient (UPG) method with momentum acceleration was proposed for solving the smooth but possibly nonconvex subproblem. This UPG method guarantees global convergence and will automatically reduce to an optimal accelerated gradient method when the smooth function in the objective is convex. Our numerical experiments on solving nonconvex quadratic programming problems and sparse optimization problems from statistical learning show that the proposed I-ADMM is very effective compared with other state-of-the-art algorithms in the literature.

关键词： Nonconvex optimization Nonsmooth optimization Separable structure Lipschitz continuous Inexact ADMM accelerated gradient method Global convergence Linear convergence rate

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Inertial projected gradient method for large-scale topology optimization

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JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2023年第2期40卷 877-905页

作者： Nishioka, Akatsuki Kanno, Yoshihiro Univ Tokyo Grad Sch Informat Sci & Technol Dept Math Informat Hongo 7-3-1Bunkyo ku Tokyo 1138656 Japan Univ Tokyo Math & Informat Ctr Hongo 7-3-1Bunkyo ku Tokyo 1138656 Japan

We present an inertial projected gradient method for solving large-scale topology optimization problems. We consider the compliance minimization problem, the heat conduction problem and the compliant mechanism problem of continua. We use the projected gradient method to efficiently treat the linear constraints of these problems. Also, inertial techniques are used to accelerate the convergence of the method. We consider an adaptive step size policy to further reduce the computational cost. The proposed method has a global convergence property. By numerical experiments, we show that the proposed method converges fast to a point satisfying the first-order optimality condition with high accuracy compared with the existing methods. The proposed method has a low computational cost per iteration, and is thus effective in a large-scale problem.

关键词： Topology optimization First-order optimization method Projected gradient method Inertial algorithm accelerated gradient method

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accelerated gradient methods with absolute and relative noise in the gradient

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OPTIMIZATION methodS & SOFTWARE 2023年第6期38卷 1180-1229页

作者： Vasin, Artem Gasnikov, Alexander Dvurechensky, Pavel Spokoiny, Vladimir Moscow Inst Phys & Technol Res Ctr Appl AI Syst Dolgoprudnyi Russia Moscow Inst Phys & Technol Lab Math Methods Optimizat Dolgoprudnyi Russia Inst Informat Transmiss Problems Sect 7 Moscow Russia Weierstrass Inst Appl Anal & Stochast Res Grp Stochast Algorithms & Nonparametr Stat Berlin Germany Humboldt Univ Inst Math Berlin Germany Weierstrass Inst Appl Anal & Stochast Res Grp Stochast Algorithms & Nonparametr Stat D-10117 Berlin Germany

In this paper, we investigate accelerated first-order methods for smooth convex optimization problems under inexact information on the gradient of the objective. The noise in the gradient is considered to be additive with two possibilities: absolute noise bounded by a constant, and relative noise proportional to the norm of the gradient. We investigate the accumulation of the errors in the convex and strongly convex settings with the main difference with most of the previous works being that the feasible set can be unbounded. The key to the latter is to prove a bound on the trajectory of the algorithm. We also give a stopping criterion for the algorithm and consider extensions to the cases of stochastic optimization and composite nonsmooth problems.

关键词： Inexact oracle accelerated gradient method convergence rate convex optimization

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Almost Sure Convergence of Proximal Stochastic accelerated gradient methods

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Journal of Applied Mathematics and Physics 2024年第4期12卷 1321-1336页

作者： Xin Xiang Haoming Xia Key Laboratory of Optimization Theory and Applications School of Mathematics and Information China West Normal University Nanchong China

Proximal gradient descent and its accelerated version are resultful methods for solving the sum of smooth and non-smooth problems. When the smooth function can be represented as a sum of multiple functions, the stochastic proximal gradient method performs well. However, research on its accelerated version remains unclear. This paper proposes a proximal stochastic accelerated gradient (PSAG) method to address problems involving a combination of smooth and non-smooth components, where the smooth part corresponds to the average of multiple block sums. Simultaneously, most of convergence analyses hold in expectation. To this end, under some mind conditions, we present an almost sure convergence of unbiased gradient estimation in the non-smooth setting. Moreover, we establish that the minimum of the squared gradient mapping norm arbitrarily converges to zero with probability one.

关键词： Proximal Stochastic accelerated method Almost Sure Convergence Composite Optimization Non-Smooth Optimization Stochastic Optimization accelerated gradient method

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Scalable Distributed Optimization Combining Conic Projection and Linear Programming for Energy Community Scheduling

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Journal of Modern Power Systems and Clean Energy 2023年第6期11卷 1814-1826页

作者： Mohammad Dolatabadi Alberto Borghetti Pierluigi Siano Department of Mathematics Vali-e-Asr University of RafsanjanRafsanjan 77188-97111Iran Department of Electrical Electronicand Information EngineeringUniversity of BolognaBolognaItaly Department of Management&Innovation Systems University of SalernoSalernoItaly Department of Electrical and Electronic Engineering Science University of JohannesburgJohannesburg 2006South Africa

In this paper, a new method to address the scheduling problem of a renewable energy community while considering network constraints and users' privacy preservation is proposed. The method decouples the optimization solution into two interacting procedures: conic projection(CP) and linear programming(LP) optimization. A new optimal CP method is proposed based on local computations and on the calculation of the roots of a fourth-order polynomial for which a closed-form solution is known. Computational tests conducted on both 14-bus and 84-bus distribution networks demonstrate the effectiveness of the proposed method in obtaining the same quality of solutions compared with that by a centralized solver. The proposed method is scalable and has features that can be implemented on microcontrollers since both LP and CP procedures require only simple matrix-vector multiplications.

关键词： accelerated gradient method battery storage system conic projection energy community energy scheduling linear programming renewable resource

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LINEAR CONVERGENCE OF PROXIMAL gradient ALGORITHM WITH EXTRAPOLATION FOR A CLASS OF NONCONVEX NONSMOOTH MINIMIZATION PROBLEMS

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SIAM JOURNAL ON OPTIMIZATION 2017年第1期27卷 124-145页

作者： Wen, Bo Chen, Xiaojun Pong, Ting Kei Harbin Inst Technol Dept Math Harbin Peoples R China Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China

In this paper, we study the proximal gradient algorithm with extrapolation for minimizing the sum of a Lipschitz differentiable function and a proper closed convex function. Under the error bound condition used in [Ann. Oper. Res., 46 (1993), pp. 157-178] for analyzing the convergence of the proximal gradient algorithm, we show that there exists a threshold such that if the extrapolation coefficients are chosen below this threshold, then the sequence generated converges R-linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also R-linearly convergent. In addition, the threshold reduces to 1 for convex problems, and as a consequence we obtain the R-linear convergence of the sequence generated by FISTA with fixed restart. Finally, we present some numerical experiments to illustrate our results.

关键词： linear convergence extrapolation error bound accelerated gradient method nonconvex nonsmooth minimization convex minimization

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accelerated Distributed Hybrid Stochastic/Robust Energy Management of Smart Grids

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IEEE TRANSACTIONS ON INDUSTRIAL INFORMATICS 2021年第8期17卷 5335-5347页

作者： Chang, Xinyue Xu, Yinliang Gu, Wei Sun, Hongbin Chow, Mo-Yuen Yi, Zhongkai Tsinghua Univ Tsinghua Shenzhen Int Grad Sch Tsinghua Berkeley Shenzhen Inst Beijing 100084 Peoples R China Southeast Univ Sch Elect Engn Nanjing 210096 Peoples R China Tsinghua Univ Dept Elect Engn State Key Lab Power Syst Beijing 100084 Peoples R China North Carolina State Univ Dept Elect & Comp Engn Raleigh NC 27695 USA

The uncertainties of renewable energy, loads, and electricity prices pose significant challenges to the economical and secure energy management of smart grids. In this article, a hybrid stochastic/robust (HSR) optimization method is developed to minimize the overall cost of all units. The proposed approach takes advantage of stochastic programming, robust optimization, and distributed optimization methods while considering various system constraints. First, stochastic electricity price scenarios are selected by the Latin hypercube sampling method. Second, the uncertainties of renewable energy generation and loads are managed by the proposed robust optimization method under each price scenario. Then, an improved distributed optimization method is proposed to solve the formulated HSR optimization problem, which considerably enhances the convergence with the accelerated gradient method. Numerical case studies of both small-scale and large-scale power systems demonstrate the accuracy, effectiveness, and scalability of the proposed distributed HSR approach. Additionally, the optimality and convergence of this proposed distributed algorithm are mathematically proven and analyzed.

关键词： Optimization methods Robustness Renewable energy sources Stochastic processes Smart grids Programming accelerated gradient method distributed optimization energy management hybrid stochastic robust (HSR) optimization smart grid

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