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A New Multi-point Stepsize Gradient Method for Optimization 5th

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A New Multi-point Stepsize Gradient Method for Optimization

5th International Conference on Numerical Analysis and Optimization: Theory, Methods, Applications and Technology Transfer, NAOV 2020

作者： Huang, Yakui Dai, Yu-Hong Liu, Xin-Wei Institute of Mathematics Hebei University of Technology Tianjin300401 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

ISBN: (纸本)9783030720391

The Barzilai-Borwein (BB) gradient method, which employs two-point stepsizes computed by the information of two consecutive iterations, is efficient in solving large-scale unconstrained optimization. In this paper, motivated by the success of BB method and the multi-point stepsize proposed by Dai and Fletcher (mathematical Programming, 2006, 106: 403–421), we develop a new efficient gradient method, called MPSG, which adaptively uses the Dai-Fletcher long stepsize and Dai-Fletcher short stepsize. The R-linear convergence of MPSG for general n-dimensional strictly convex quadratic functions is established. By making use of two modified multi-point stepsizes and nonmonotone line searches, MPSG is extended to solve general unconstrained optimization. Moreover, the proposed MPSG method is further generalized for solving extreme eigenvalue problems. Numerical experiments on quadratic optimization, general unconstrained optimization and extreme eigenvalue problems demonstrate the efficiency of our method. © 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

关键词： Gradient methods

Asymptotic SEP Analysis and Optimization of Linear-Quantized Precoding in Massive MIMO Systems

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arXiv 2023年

作者： Wu, Zheyu Ma, Junjie Liu, Ya-Feng Swindlehurst, A. Lee The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China The Center for Pervasive Communications and Computing University of California IrvineCA92697 United States

A promising approach to deal with the high hardware cost and energy consumption of massive MIMO transmitters is to use low-resolution digital-to-analog converters (DACs) at each antenna element. This leads to a transmission scheme where the transmitted signals are restricted to a finite set of voltage levels. This paper is concerned with the analysis and optimization of a low-cost quantized precoding strategy, referred to as linear-quantized precoding, for a downlink massive MIMO system under Rayleigh fading. In linear-quantized precoding, the signals are first processed by a linear precoding matrix and subsequently quantized component-wise by the DAC. In this paper, we analyze both the signal-to-interference-plus-noise ratio (SINR) and the symbol error probability (SEP) performances of such linear-quantized precoding schemes in an asymptotic framework where the number of transmit antennas and the number of users grow large with a fixed ratio. Our results provide a rigorous justification for the heuristic arguments based on the Bussgang decomposition that are commonly used in prior works. Based on the asymptotic analysis, we further derive the optimal precoder within a class of linear-quantized precoders that includes several popular precoders as special cases. Our numerical results demonstrate the excellent accuracy of the asymptotic analysis for finite systems and the optimality of the derived precoder. Copyright © 2023, The Authors. All rights reserved.

关键词： MIMO systems

A Dynamical Variable-separation Method for Parameter-dependent Dynamical Systems

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arXiv 2025年

作者： Chen, Liang Chen, Yaru Li, Qiuqi Zhou, Tao School of Mathematics Hunan University Changsha410082 China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This paper proposes a dynamical Variable-separation method for solving parameter-dependent dynamical systems. To achieve this, we establish a dynamical low-rank approximation for the solutions of these dynamical systems by successively enriching each term in the reduced basis functions via a greedy algorithm. This enables us to reformulate the problem to two decoupled evolution equations at each enrichment step, of which one is a parameter-independent partial differential equation and the other one is a parameter-dependent ordinary differential equation. These equations are directly derived from the original dynamical system and the previously separate representation terms. Moreover, the computational process of the proposed method can be split into an offline stage, in which the reduced basis functions are constructed, and an online stage, in which the efficient low-rank representation of the solution is employed. The proposed dynamical Variable-separation method is capable of offering reduced computational complexity and enhanced efficiency compared to many existing low-rank separation techniques. Finally, we present various numerical results for linear/nonlinear parameter-dependent dynamical systems to demonstrate the effectiveness of the proposed *** Codes 35R60, 37M99, 65P99 Copyright © 2025, The Authors. All rights reserved.

关键词： Ordinary differential equations

Detecting and Exploiting Unit Commitment Structures in MIPs 6

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Detecting and Exploiting Unit Commitment Structures in MIPs

6th International Conference on Power and Energy Technology, ICPET 2024

作者： Su, Zhao-Gang Chen, Sheng-Jie Li, Yu-Fei Chen, Liang Kou, Caixia Cai, Zhi Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Institute of Applied Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China College of Sciences Beijing University of Posts and Telecommunications Beijing China Beijing Key Laboratory of Research and System Evaluation of Power Dispatching Automation Technology China Electric Power Research Institute Beijing China

ISBN: (纸本)9798350375855

The unit commitment (UC) problem has been extensively researched in the literature, which is typically formulated as a mixed integer programming (MIP) problem. However, current studies lack effective methods to identify potential UC problem structures within a general MIP problem, making it challenging to leverage well-established valid inequalities for UC problems to accelerate the solution process. In this paper, we present a novel detection method that automatically identifies UC structures within an MIP problem by analyzing the relationships between variables and constraints. Moreover, we enhance the MIP formulation by incorporating three well-studied classes of valid inequalities as cutting planes, based on the identified UC structures. Our experiments demonstrate that our proposed UC detection method greatly improves computational efficacy of a standard MIP solver compared to the default setting. © 2024 IEEE.

关键词： Mixed-integer linear programming

A New Function Space from Barron Class and Application to Neural Network Approximation

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Communications in computational Physics 2022年第10期32卷 1361-1400页

作者： Yan Meng Pingbing Ming School of Mathematics Renmin University of ChinaBeijing 100872China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesNo.55East Road Zhong-Guan-CunBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China.

We introduce a new function space,dubbed as the Barron spectrum space,which arises from the target function space for the neural network *** give a Bernstein type sufficient condition for functions in this space,and clarify the embedding among the Barron spectrum space,the Bessel potential space,the Besov space and the Sobolev ***,the unexpected smoothness and the decaying behavior of the radial functions in the Barron spectrum space have been *** an application,we prove a dimension explicit L^(q) error bound for the two-layer neural network with the Barron spectrum space as the target function space,the rate is dimension independent.

关键词： Fourier transform Besov space Sobolev space radial function neural network

A Symplectic Discretization Based Proximal Point Algorithm for Convex Minimization

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arXiv 2024年

作者： Yuan, Ya-Xiang Zhang, Yi Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China University of Chinese Academy of Sciences Beijing100049 China

The proximal point algorithm plays a central role in non-smooth convex programming. The Augmented Lagrangian Method, one of the most famous optimization algorithms, has been found to be closely related to the proximal point algorithm. Due to its importance, accelerated variants of the proximal point algorithm have received considerable attention. In this paper, we first study an Ordinary Differential Equation (ODE) system, which provides valuable insights into proving the convergence rate of the desired algorithm. Using the Lyapunov function technique, we establish the convergence rate of the ODE system. Next, we apply the Symplectic Euler Method to discretize the ODE system to derive a new proximal point algorithm, called the Symplectic Proximal Point Algorithm (SPPA). By utilizing the proof techniques developed for the ODE system, we demonstrate the convergence rate of the SPPA. Additionally, it is shown that existing accelerated proximal point algorithm can be considered a special case of the SPPA in a specific manner. Furthermore, under several additional assumptions, we prove that the SPPA exhibits a finer convergence *** Codes 49J52, 65B99, 65K10, 65K15, 68Q25, 90C25, 90C26 © 2024, CC BY.

关键词： Lyapunov functions

SYMPLECTIC EXTRA-GRADIENT TYPE METHOD FOR SOLVING GENERAL NON-MONOTONE INCLUSION PROBLEM

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arXiv 2024年

In recent years, accelerated extra-gradient methods have attracted much attention by researchers, for solving monotone inclusion problems. A limitation of most current accelerated extra-gradient methods lies in their direct utilization of the initial point, which can potentially decelerate numerical convergence rate. In this work, we present a new accelerated extra-gradient method, by utilizing the symplectic acceleration technique. We establish the inverse of quadratic convergence rate by employing the Lyapunov function technique. Also, we demonstrate a faster inverse of quadratic convergence rate alongside its weak convergence property under stronger assumptions. To improve practical efficiency, we introduce a line search technique for our symplectic extra-gradient method. Theoretically, we prove the convergence of the symplectic extra-gradient method with line search. Numerical tests show that this adaptation exhibits faster convergence rates in practice compared to several existing extra-gradient type *** Codes 47J20, 47H05, 65K10, 65K15, 65Y20, 90C30, 90C52 © 2024, CC BY.

关键词： Lyapunov functions

QoS-Aware and Routing-Flexible Network Slicing for Service-Oriented Networks

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arXiv 2024年

作者： Chen, Wei-Kun Liu, Ya-Feng Dai, Yu-Hong Luo, Zhi-Quan The School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Beijing100081 China The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China The Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen518172 China

In this paper, we consider the network slicing (NS) problem which attempts to map multiple customized virtual network requests (also called services) to a common shared network infrastructure and manage network resources to meet diverse quality of service (QoS) requirements. We propose a mixed-integer nonlinear programming (MINLP) formulation for the considered NS problem that can flexibly route the traffic flow of the services on multiple paths and provide end-to-end delay and reliability guarantees for all services. To overcome the computational difficulty due to the intrinsic nonlinearity in the MINLP formulation, we transform the MINLP formulation into an equivalent mixed-integer linear programming (MILP) formulation and further show that their continuous relaxations are equivalent. In sharp contrast to the continuous relaxation of the MINLP formulation which is a nonconvex nonlinear programming problem, the continuous relaxation of the MILP formulation is a polynomial-time solvable linear programming problem, which significantly facilitates the algorithmic design. Based on the newly proposed MILP formulation, we develop a customized column generation (cCG) algorithm for solving the NS problem. The proposed cCG algorithm is a decomposition-based algorithm and is particularly suitable for solving large-scale NS problems. Numerical results demonstrate the efficacy of the proposed formulations and the proposed cCG algorithm. Copyright © 2024, The Authors. All rights reserved.

关键词： Mixed-integer linear programming

Asymptotic Estimates for Spectral Estimators of Rotationally Invariant Matrices

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Asymptotic Estimates for Spectral Estimators of Rotationally...

IEEE International Symposium on Information Theory

作者： Zhuohang He Xiaojun Yuan Junjie Ma National Key Laboratory of Wireless Communications University of Electronic Science and Technology of China Chengdu China Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences Beijing China

ISBN: (数字)9798350382846

ISBN: (纸本)9798350382853

In this paper, we consider the recovery of low-rank matrices from noisy observations using spectral denoisers, where the singular values are denoised through an identical scalar smoothing function. We explore the asymptotic mean squared error (AMSE) of these denoisers within a framework where the rank of the matrix to be recovered grows linearly with the matrix size. We demonstrate that, under arbitrary i.i.d. noise and some mild regularity assumptions, the AMSE converges in probability to a deterministic function of the noise power. Our results are applicable to commonly used denoisers, including the best-rank-r denoiser, the singular-value soft-threshold denoiser, and the singular-value hard-threshold denoiser. To the best of our knowledge, this is the first study to establish an analytical expression for the asymptotic MSE under arbitrary i.i.d. noise. The derived analytical expression depends solely on the empirical distribution of the singular values of the low-rank matrix and the specific form of the spectral denoiser employed.

关键词： Smoothing methods Limiting Noise Noise reduction Noise measurement Information theory