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arXiv 2024年

作者： Hu, Yukuan Yin, Junlei Gao, Xingyu Liu, Xin Song, Haifeng 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 School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China State Key Laboratory of Solidification Processing Northwestern Polytechnical University Shaanxi Xi’an710072 China Innovation Center NPU Chongqing Chongqing401135 China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing100088 China

This paper is concerned with ab initio crystal structure relaxation under a fixed unit cell volume, which is a step in calculating the static equations of state and forms the basis of thermodynamic property calculations for materials. The task can be formulated as an energy minimization with a determinant constraint. Widely used line minimization-based methods (e.g., conjugate gradient method) lack both efficiency and convergence guarantees due to the nonconvex nature of the feasible region as well as the significant differences in the curvatures of the potential energy surface with respect to atomic and lattice components. To this end, we propose a projected gradient descent algorithm named PANBB. It is equipped with (i) search direction projections onto the tangent spaces of the nonconvex feasible region for lattice vectors, (ii) distinct curvature-aware initial trial step sizes for atomic and lattice updates, and (iii) a nonrestrictive line minimization criterion as the stopping rule for the inner loop. It can be proved that PANBB favors theoretical convergence to equilibrium states. Across a benchmark set containing 223 structures from various categories, PANBB achieves average speedup factors of approximately 1.41 and 1.45 over the conjugate gradient method and direct inversion in the iterative subspace implemented in off-the-shelf simulation software, respectively. Moreover, it normally converges on all the systems, manifesting its unparalleled robustness. As an application, we calculate the static equations of state for the high-entropy alloy AlCoCrFeNi, which remains elusive owing to 160 atoms representing both chemical and magnetic disorder and the strong local lattice distortion. The results are consistent with the previous calculations and are further validated by experimental thermodynamic data. © 2024, CC BY.

关键词： Conjugate gradient method

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A New Adaptive Balanced Augmented Lagrangian Method with Application to ISAC Beamforming Design

arXiv

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arXiv 2024年

作者： Wu, Jiageng Jiang, Bo Li, Xinxin Liu, Ya-Feng Yuan, Jianhua School of Mathematics Jilin University Changchun130012 China Ministry of Education Key Laboratory for NSLSCS School of Mathematical Sciences Nanjing Normal University Nanjing210023 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 School of Sciences Beijing University of Posts and Telecommunications Beijing100876 China

In this paper, we consider a class of convex programming problems with linear equality constraints, which finds broad applications in machine learning and signal processing. We propose a new adaptive balanced augmented Lagrangian (ABAL) method for solving these problems. The proposed ABAL method adaptively selects the stepsize parameter and enjoys a low per-iteration complexity, involving only the computation of a proximal mapping of the objective function and the solution of a linear equation. These features make the proposed method wellsuited to large-scale problems. We then custom-apply the ABAL method to solve the ISAC beamforming design problem, which is formulated as a nonlinear semidefinite program in a previous work. This customized application requires careful exploitation of the problem's special structure such as the property that all of its signal-to-interference-and-noise-ratio (SINR) constraints hold with equality at the solution and an efficient computation of the proximal mapping of the objective function. Simulation results demonstrate the efficiency of the proposed ABAL method. © 2024, CC BY.

关键词： Signal to noise ratio

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Stochastic optimization over expectation-formulated generalized Stiefel manifold

arXiv

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arXiv 2024年

作者： Jiang, Linshuo Xiao, Nachuan Liu, Xin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China The Institute of Operations Research and Analytics National University of Singapore Singapore

In this paper, we consider a class of stochastic optimization problems over the expectation-formulated generalized Stiefel manifold (SOEGS), where the objective function f is continuously differentiable. We propose a novel constraint dissolving penalty function with a customized penalty term (CDFCP), which maintains the same order of differentiability as f. Our theoretical analysis establishes the global equivalence between CDFCP and SOEGS, in the sense that they share the same first-order and second-order stationary points under mild conditions. These results on equivalence enable the direct implementation of various stochastic optimization approaches to solve SOEGS. In particular, we develop a stochastic gradient algorithm and its accelerated variant by incorporating an adaptive step size strategy. Furthermore, we prove their O(Ε−4) sample complexity for finding an Ε-stationary point of CDFCP. Comprehensive numerical experiments show the efficiency and robustness of our proposed algorithms. Copyright © 2024, The Authors. All rights reserved.

关键词： Optimization algorithms

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Decentralized Stochastic Subgradient-based Methods for Nonsmooth Nonconvex Optimization

arXiv

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arXiv 2024年

作者： Zhang, Siyuan Xiao, Nachuan Liu, Xin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences China The Institute of Operations Research and Analytics National University of Singapore Singapore

In this paper, we concentrate on decentralized optimization problems with nonconvex and nonsmooth objective functions, especially on the decentralized training of nonsmooth neural networks. We introduce a unified framework to analyze the global convergence of decentralized stochastic subgradient-based methods. We prove the global convergence of our proposed framework under mild conditions, by establishing that the generated sequence asymptotically approximates the trajectories of its associated differential inclusion. Furthermore, we establish that our proposed framework covers a wide range of existing efficient decentralized subgradient-based methods, including decentralized stochastic subgradient descent (DSGD), DSGD with gradient-tracking technique (DSGD-T), and DSGD with momentum (DSGD-M). In addition, we introduce the sign map to regularize the update directions in DSGD-M, and show it is enclosed in our proposed framework. Consequently, our convergence results establish, for the first time, global convergence of these methods when applied to nonsmooth nonconvex objectives. Preliminary numerical experiments demonstrate that our proposed framework yields highly efficient decentralized subgradient-based methods with convergence guarantees in the training of nonsmooth neural networks. Copyright © 2024, The Authors. All rights reserved.

关键词： Lyapunov functions

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Numerical Solution for Nonlinear 4D Variational Data Assimilation (4D-Var) via ADMM

arXiv

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arXiv 2024年

作者： Li, Bowen Shi, Bin State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

The four-dimensional variational data assimilation (4D-Var) has emerged as an important methodology, widely used in numerical weather prediction, oceanographic modeling, and climate forecasting. Classical unconstrained gradient-based algorithms often struggle with local minima, making their numerical performance highly sensitive to the initial guess. In this study, we exploit the separable structure of the 4D-Var problem to propose a practical variant of the alternating direction method of multipliers (ADMM), referred to as the linearized multi-block ADMM with regularization. Unlike classical first-order optimization methods that primarily focus on initial conditions, our approach derives the Euler-Lagrange equation for the entire dynamical system, enabling more comprehensive and effective utilization of observational data. When the initial condition is poorly chosen, the arg min operation steers the iteration towards the observational data, thereby reducing sensitivity to the initial guess. The quadratic subproblems further simplify the solution process, while the parallel structure enhances computational efficiency, especially when utilizing modern hardware. To validate our approach, we demonstrate its superior performance using the Lorenz system, even in the presence of noisy observational data. Furthermore, we showcase the effectiveness of the linearized multi-block ADMM with regularization in solving the 4D-Var problems for the viscous Burgers’ equation, across various numerical schemes, including finite difference, finite element, and spectral methods. Finally, we illustrate the recovery of dynamics under noisy observational data in a 2D turbulence scenario, particularly focusing on vorticity concentration, highlighting the robustness of our algorithm in handling complex physical phenomena. © 2024, CC BY-NC-ND.

关键词： Weather forecasting

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A Decentralized Proximal Gradient Tracking Algorithm for Composite Optimization on Riemannian Manifolds

arXiv

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arXiv 2024年

作者： Wang, Lei Bao, Le Liu, Xin Department of Statistics Pennsylvania State University University ParkPA United States State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences Beijing China

This paper focuses on minimizing a smooth function combined with a nonsmooth regularization term on a compact Riemannian submanifold embedded in the Euclidean space under a decentralized setting. Typically, there are two types of approaches at present for tackling such composite optimization problems. The first, subgradient-based approaches, rely on subgradient information of the objective function to update variables, achieving an iteration complexity of O(ϵ-4 log2(ϵ-2)). The second, smoothing approaches, involve constructing a smooth approximation of the nonsmooth regularization term, resulting in an iteration complexity of O(ϵ-4). This paper proposes a proximal gradient type algorithm that fully exploits the composite structure. The global convergence to a stationary point is established with a significantly improved iteration complexity of O(ϵ-2). To validate the effectiveness and efficiency of our proposed method, we present numerical results in real-world applications, showcasing its superior performance. Copyright © 2024, The Authors. All rights reserved.

关键词： Iterative methods

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The Distributionally Robust Optimization Model of Sparse Principal Component Analysis

arXiv

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arXiv 2025年

作者： Wang, Lei Liu, Xin Chen, Xiaojun Department of Applied Mathematics The Hong Kong Polytechnic University Hong Kong State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences University of Chinese Academy of Sciences Beijing China

We consider sparse principal component analysis (PCA) under a stochastic setting where the underlying probability distribution of the random parameter is uncertain. This problem is formulated as a distributionally robust optimization (DRO) model based on a constructive approach to capturing uncertainty in the covariance matrix, which constitutes a nonsmooth constrained min-max optimization problem. We further prove that the inner maximization problem admits a closed-form solution, reformulating the original DRO model into an equivalent minimization problem on the Stiefel manifold. This transformation leads to a Riemannian optimization problem with intricate nonsmooth terms, a challenging formulation beyond the reach of existing algorithms. To address this issue, we devise an efficient smoothing manifold proximal gradient algorithm. We prove the Riemannian gradient consistency and global convergence of our algorithm to a stationary point of the nonsmooth minimization problem. Moreover, we establish the iteration complexity of our algorithm. Finally, numerical experiments are conducted to validate the effectiveness and scalability of our algorithm, as well as to highlight the necessity and rationality of adopting the DRO model for sparse PCA. Copyright © 2025, The Authors. All rights reserved.

关键词： Optimization algorithms

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Unconditionally optimal convergence of an energy-conserving and linearly implicit scheme for nonlinear wave equations

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Science China Mathematics 2022年第8期65卷 1731-1748页

作者： Waixiang Cao Dongfang Li Zhimin Zhang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing 100193China Department of Mathematics Wayne State UniversityDetroitMI 48202USA

In this paper,we present and analyze an energy-conserving and linearly implicit scheme for solving the nonlinear wave *** error estimates in time and superconvergent error estimates in space are established without certain time-step *** key is to estimate directly the solution bounds in the H-norm for both the nonlinear wave equation and the corresponding fully discrete scheme,while the previous investigations rely on the temporal-spatial error splitting *** examples are presented to confirm energy-conserving properties,unconditional convergence and optimal error estimates,respectively,of the proposed fully discrete schemes.

关键词： scalar auxiliary variable wave equations stability error estimate superconvergence

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Optimal-rate error estimates and a twice decoupled solver for a backward Euler finite element scheme of the Doyle-Fuller-Newman model of lithium-ion batteries

arXiv

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arXiv 2024年

作者： Xu, Shu Cao, Liqun School of Mathematical Sciences Peking University Beijing100871 China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China State Key Laboratory of Scientific and Engineering Computing Beijing100190 China National Center for Mathematics and Interdisciplinary Sciences Chinese Academy of Sciences Beijing100190 China

We investigate the convergence of a backward Euler finite element discretization applied to a multi-domain and multi-scale elliptic-parabolic system, derived from the Doyle-Fuller-Newman model for lithium-ion batteries. We establish optimal-order error estimates for the solution in the norms l2(H1) and l2(L2(Hrq)), q = 0,1. To improve computational efficiency, we propose a novel solver that accelerates the solution process and controls memory usage. Numerical experiments with realistic battery parameters validate the theoretical error rates and demonstrate the significantly superior performance of the proposed solver over existing solvers. Copyright © 2024, The Authors. All rights reserved.

关键词： Computational efficiency

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Adam-family methods for nonsmooth optimization with convergence guarantees

The Journal of Machine Learning Research

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The Journal of Machine Learning Research 2024年第1期25卷 2398-2450页

作者： Nachuan Xiao Xiaoyin Hu Xin Liu Kim-Chuan Toh Institute of Operations Research and Analytics National University of Singapore Singapore School of Computer and Computing Science Hangzhou City University Hangzhou China State Key Laboratory of Scientific and Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Department of Mathematics and Institute of Operations Research and Analytics National University of Singapore Singapore

In this paper, we present a comprehensive study on the convergence properties of Adam-family methods for nonsmooth optimization, especially in the training of nonsmooth neural networks. We introduce a novel two-timescale framework that adopts a two-timescale updating scheme, and prove its convergence properties under mild assumptions. Our proposed framework encompasses various popular Adam-family methods, providing convergence guarantees for these methods in training nonsmooth neural networks. Furthermore, we develop stochastic subgradient methods that incorporate gradient clipping techniques for training nonsmooth neural networks with heavy-tailed noise. Through our framework, we show that our proposed methods converge even when the evaluation noises are only assumed to be integrable. Extensive numerical experiments demonstrate the high efficiency and robustness of our proposed methods.

关键词： nonsmooth optimization stochastic subgradient methods Adam nonconvex optimization gradient clipping

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