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Inertial projected gradient method for large-scale topology optimization

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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Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope

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OPTICAL ENGINEERING 2016年第3期55卷 1-10页

作者： Wang, Hairen Lou, Zheng Qian, Yuan Zheng, Xianzhong Zuo, Yingxi Chinese Acad Sci Purple Mt Observ 2 West Beijing Rd Nanjing 210008 Jiangsu Peoples R China Chinese Acad Sci Key Lab Radio Astron 2 West Beijing Rd Nanjing 210008 Jiangsu Peoples R China

The optimization of a primary mirror support system is one of the most critical problems in the design of large telescopes. Here, we propose a hybrid optimization methodology of variable densities mesh model (HOMVDMM) for the axial supporting design, which has three key steps: (1) creating a variable densities mesh model, which will partition the mirror into several sparse mesh areas and several dense mesh areas;(2) global optimization based on the zero-order optimization method for the support of primary mirror with a large tolerance;(3) based on the optimization results of the second step, further optimization with first-order optimization method in dense mesh areas by a small tolerance. HOMVDMM exploits the complementary merits of both the zero-and first-order optimizations, with the former in global scale and the latter in small scale. As an application, the axial support of the primary mirror of the 2.5-m wide-field survey telescope (WFST) is optimized by HOMVDMM. These three designs are obtained via a comparative study of different supporting points including 27 supporting points, 39 supporting points, and 54 supporting points. Their residual half-path length errors are 28.78, 9.32, and 5.29 nm. The latter two designs both meet the specification of WFST. In each of the three designs, a global optimization value with high accuracy will be obtained in an hour on an ordinary PC. As the results suggest, the overall performance of HOMVDMM is superior to the first-order optimization method as well as the zero-order optimization method. (C) 2016 Society of Photo-Optical Instrumentation Engineers (SPIE)

关键词： hybrid optimization methodology variable densities mesh model axial support optimization primary mirror zero-order optimization method first-order optimization method

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Steel truss bridges with welded box-section members and bowknot integral joints, Part II: Minimum weight optimization

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JOURNAL OF CONSTRUCTIONAL STEEL RESEARCH 2013年 80卷 475-482页

作者： Cheng, Bin Qian, Qin Sun, Haitao Shanghai Jiao Tong Univ Dept Civil Engn Shanghai 200240 Peoples R China Shanghai Municipal Engn Design Gen Inst Shanghai 200092 Peoples R China

The second part of the paper presents the minimum weight optimization based on the provisions of current design codes for both conventional and bowknot trusses. The optimization philosophies are illustrated first through the analytic derivation of minimum weight optimization of a single member. These results indicate that the member weight reduction increases as the primary stress to secondary stress ratio, or the end moment reduction, is increased. The minimum weight optimization of the truss then proceeds on the basis of linear finite element analysis of the same truss that was discussed in Part I, by the use of first-order optimization method. In the optimization formulation, the cost rise due to steel strength enhancement of shrunken segments is taken into account in the nominal weight of whole truss, and a series of requirements related to truss vertical stiffness, member strength, member stability, and truss stability are set as constraint functions. The optimization results show that the optimal weights of bowknot trusses are less than those of conventional trusses on the premise that the steel strength of shrunken segments has been moderately improved. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Steel truss Bowknot integral joint Finite element method first-order optimization method Minimum weight optimization

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Designing Universally-Approximating Deep Neural Networks: A first-order optimization Approach

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 2024年第9期46卷 6231-6246页

作者： Wu, Zhoutong Xiao, Mingqing Fang, Cong Lin, Zhouchen Peking Univ Acad Adv Interdisciplinary Studies Ctr Data Sci Beijing 100871 Peoples R China Peking Univ Sch Intelligence Sci & Technol Natl Key Lab Gen AI Beijing 100871 Peoples R China Peking Univ Inst Artificial Intelligence Sch Intelligence Sci & Technol Natl Key Lab Gen AI Beijing 100871 Peoples R China Peng Cheng Lab Guangzhou 518066 Peoples R China

Universal approximation capability, also referred to as universality, is an important property of deep neural networks, endowing them with the potency to accurately represent the underlying target function in learning tasks. In practice, the architecture of deep neural networks largely influences the performance of the models. However, most existing methodologies for designing neural architectures, such as the heuristic manual design or neural architecture search, ignore the universal approximation property, thus losing a potential safeguard about the performance. In this paper, we propose a unified framework to design the architectures of deep neural networks with a universality guarantee based on first-order optimization algorithms, where the forward pass is interpreted as the updates of an optimization algorithm. The (explicit or implicit) network is designed by replacing each gradient term in the algorithm with a learnable module similar to a two-layer network or its derivatives. Specifically, we explore the realm of width-bounded neural networks, a common practical scenario, showcasing their universality. Moreover, adding operations of normalization, downsampling, and upsampling does not hurt the universality. To the best of our knowledge, this is the first work that width-bounded networks with universal approximation guarantee can be designed in a principled way. Our framework can inspire a variety of neural architectures including some renowned structures such as ResNet and DenseNet, as well as novel innovations. The experimental results on image classification problems demonstrate that the newly inspired networks are competitive and surpass the baselines of ResNet, DenseNet, as well as the advanced ConvNeXt and ViT, testifying to the effectiveness of our framework.

关键词： optimization Approximation algorithms Design methodology Manuals Reviews Knowledge engineering Image classification Neural network design universal approximation first-order optimization method skip connection width-bounded

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Accelerated proximal gradient method for bi-modulus static elasticity

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optimization AND ENGINEERING 2022年第1期23卷 453-477页

作者： Kanno, Yoshihiro Univ Tokyo Math & Informat Ctr Hongo 7-3-1 Tokyo 1138656 Japan

Bi-modulus constitutive law assumes that material constants have different values in tension and compression. It is known that finding an equilibrium state of an elastic body consisting of a bi-modulus material is recast as a semidefinite programming problem, which can be solved with a primal-dual interior-point method. As an alternative approach, this paper presents a fast first-order optimization method. Specifically, we propose an accelerated proximal gradient method for solving a minimization problem of the total potential energy. This algorithm is easy to implement, and free from numerical solution of linear equations. Numerical experiments demonstrate that the proposed method outperforms the semidefinite programming approach with a standard solver implementing a primal-dual interior-point method.

关键词： Bi-modulus material Convex optimization first-order optimization method Proximal gradient method Semidefinite programming

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Dynamic energy management in smart grid: A fast randomized first-order optimization algorithm

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INTERNATIONAL JOURNAL OF ELECTRICAL POWER & ENERGY SYSTEMS 2018年 94卷 179-187页

作者： Han, Dong Sun, Weiqing Fan, Xiang Univ Shanghai Sci & Technol Dept Elect Engn Shanghai 200093 Peoples R China Guizhou Power Grid Corp Ltd Power Dispatch & Control Ctr Guiyang 550000 Guizhou Peoples R China

A crucial issue in the smart grid is how to manage the controllable load resources of end-users, in order to reduce the economic costs of system operation and facilitate to utilize renewable energies. This paper investigates a fast randomized first-order optimization method to explore the solution of dynamic energy management (DEM) for the smart grid integrated large-scale distributed energy resources. A complicated time-coupling and multi-variable optimal problem is presented to determine the load scheduling for the electricity customers. The main challenge of the proposed problem is to enable the efficient processing of the large data volumes and optimization of aggregated data involved in DEM. The first-order method as one of big data optimization algorithms is able to exhibit significant performance for computing globally optimal solutions based on randomization techniques. Using such solution approach, we can reformulate the original problem into an unconstrained augmented Lagrangian function. The optimal results can be obtained from computing the gradient based on the information of the first-order derivative. To speed up the calculations of obtaining the feasible solutions, the optimization variable matrix used to update the Lagrangian multiplier can be replaced with the corresponding low-rank representation in the iteration process. Both theoretical analysis and simulation results suggest that the proposed approach may effectively solve the optimal scheduling problem of DEM considering users' participation. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Dynamic energy management Distributed energy resources first-order optimization method Augmented Lagrangian function Low-rank matrix approximation

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A note on a family of proximal gradient methods for quasi-static incremental problems in elastoplastic analysis

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Theoretical & Applied Mechanics Letters 2020年第5期10卷 315-320页

作者： Yoshihiro Kanno Mathematics and Informatics Center University of TokyoHongo 7-3-1Tokyo 113-8656Japan

Accelerated proximal gradient methods have recently been developed for solving quasi-static incremental problems of elastoplastic analysis with some different yield *** has been demonstrated through numerical experiments that these methods can outperform conventional optimization-based approaches in computational ***,in literature these algorithms are described individually for specific yield criteria,and hence there exists no guide for application of the algorithms to other yield *** short paper presents a general form of algorithm design,independent of specific forms of yield criteria,that unifies the existing proximal gradient *** interpretation is also given to each step of the presented general algorithm so that each update rule is linked to the underlying physical laws in terms of mechanical quantities.

关键词： Elastoplastic analysis Incremental problem Nonsmooth convex optimization first-order optimization method Proximal gradient method

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Least Favorable Compressed Sensing Problems for first-order methods

Least Favorable Compressed Sensing Problems for First-Order ...

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IEEE International Symposium on Information Theory (ISIT)

作者： Maleki, Arian Baraniuk, Richard Rice Univ Dept Elect & Comp Engn Houston TX 77251 USA

ISBN: (纸本)9781457705953

Compressed sensing (CS) exploits the compressibility of natural signals to reduce the number of samples required for accurate reconstruction. The cost for sub-Nyquist sampling has been computationally expensive reconstruction algorithms, including large-scale l(1) optimization. Therefore, first-order optimization methods that exploit only the gradient of the reconstruction cost function have been developed;notable examples include iterative soft thresholding (IST), fast iterative soft thresholding algorithm (FISTA), and approximate message passing (AMP). The performance of these algorithms has been studied mainly in the standard framework of convex optimization, called the deterministic framework here. In this paper, we first show that the deterministic approach results in overly pessimistic conclusions that are not indicative of algorithm performance in practice. As an alternative to the deterministic framework, we second study the theoretical aspects of the statistical convergence rate, a topic that has remained unexplored in the sparse recovery literature. Our theoretical and empirical studies reveal several hallmark properties of the statistical convergence of first-order methods, including universality over the matrix ensemble and the least favorable coefficient distribution.

关键词： AMP FISTA IST approximate message passing convex optimization convex programming deterministic framework fast iterative soft thresholding algorithm first-order optimization method gradient methods iterative soft thresholding large-scale optimization least favorable coefficient distribution least favorable compressed sensing problem matrix ensemble reconstruction cost gradient function signal reconstruction signal sampling sparse recovery literature statistical convergence rate subNyquist sampling cost

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