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A Wolfe line search algorithm for Vector Optimization

ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2019年第4期45卷 1–23页

作者： Lucambio Perez, L. R. Prudente, L. F. Univ Fed Goias Inst Matemat & Estat Ave Esperanca S-NCampus Samambaia BR-74690900 Goiania Go Brazil

In a recent article, Lucambio Perez and Prudente extended the Wolfe conditions for the vector-valued optimization. Here, we propose a line search algorithm for finding a step size satisfying the strong Wolfe conditions in the vector optimization setting. Well definedness and finite termination results are provided. We discuss practical aspects related to the algorithm and present some numerical experiments illustrating its applicability. Codes supporting this article are written in Fortran 90 and are freely available for download.

关键词： line search algorithm Wolfe conditions vector optimization

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Nonlinear conjugate gradient method for vector optimization on Riemannian manifolds with retraction and vector transport

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APPLIED MATHEMATICS AND COMPUTATION 2025年 486卷

作者： Chen, Kangming Fukuda, Ellen Hidemi Sato, Hiroyuki Kyoto Univ Yoshida Honmachi Kyoto Kyoto 6068501 Japan Ritsumeikan Univ Noji Higashi Kusatsu Shiga 5258577 Japan

In this paper, we propose nonlinear conjugate gradient methods for vector optimization on Riemannian manifolds. The concepts of Wolfe and Zoutendjik conditions are extended to Riemannian manifolds. Specifically, the existence of intervals of step sizes that satisfy the Wolfe conditions is established. The convergence analysis covers the vector extensions of the Fletcher- Reeves, conjugate descent, and Dai-Yuan parameters. Under some assumptions, we prove that the sequence obtained by the proposed algorithm can converge to a Pareto stationary point. Moreover, several other choices of the parameter are discussed. Numerical experiments illustrating the practical behavior of the methods are presented.

关键词： Conjugate gradient method Vector optimization Riemannian manifolds Wolfe conditions line search algorithm

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A simple introduction to the SiMPL method for density-based topology optimization

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2025年第4期68卷 1-17页

作者： Kim, Dohyun Lazarov, Boyan S. Surowiec, Thomas M. Keith, Brendan Brown Univ Div Appl Math Providence RI 02912 USA Lawrence Livermore Natl Lab Livermore CA 94550 USA Simula Res Lab Dept Numer Anal & Sci Comp N-0164 Oslo Norway

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as "the simple method") optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi-Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems.

关键词： Topology optimization Projected mirror descent Numerical optimization line search algorithm

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Modification of the wolfe line search rules to satisfy the descent condition in the polak-ribiere-polyak conjugate gradient method

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2007年第2期132卷 287-305页

作者： Armand, P. Di Pillo, G. Univ Limoges F-87065 Limoges France

This paper proposes a line search technique to satisfy a relaxed form of the strong Wolfe conditions in order to guarantee the descent condition at each iteration of the Polak-Ribiere-Polyak conjugate gradient algorithm. It is proved that this line search algorithm preserves the usual convergence properties of any descent algorithm. In particular, it is shown that the Zoutendijk condition holds under mild assumptions. It is also proved that the resulting conjugate gradient algorithm is convergent under a strong convexity assumption. For the nonconvex case, a globally convergent modification is proposed. Numerical tests are presented.

关键词： conjugate gradient method line search algorithm unconstrained optimization

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Hybrid particle swarm optimizer with line search

Hybrid particle swarm optimizer with line search

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IEEE International Conference on Systems, Man and Cybernetics

作者： Liu, Y Qin, Z Shi, ZW Xi An Jiao Tong Univ Dept Comp Sci Xian 710049 Peoples R China

ISBN: (纸本)0780385667

Particle swarm optimization, a new good swarm intelligence paradigm, has been successfully applied to many non-linear optimization problems. In a swarm each particle adjusts its flying toward a promising area depending on cooperative interaction with others. The cooperative interaction of particles provides effective ways to determine the right flying direction for every particle, which is the key reason for the success of PSO. However, previous PSO algorithms are not good at choosing the step-size along the promising direction. In this paper a line search method is employed to enhance particle swarm optimizer so that the step size is chosen rationally. The experimental results show that PSO with line search method has a potential to achieve better solutions.

关键词： particle swarm optimization line search algorithm golden section algorithm

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A family of conjugate gradient methods with guaranteed positiveness and descent for vector optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2024年第3期89卷 805-842页

作者： He, Qing-Rui Li, Sheng-Jie Zhang, Bo-Ya Chen, Chun-Rong Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China Chongqing Univ Key Lab Nonlinear Anal & Its Applicat Minist Educ Chongqing Peoples R China

In this paper, we seek a new modification way to ensure the positiveness of the conjugate parameter and, based on the Dai-Yuan (DY) method in the vector setting, propose an associated family of conjugate gradient (CG) methods with guaranteed descent for solving unconstrained vector optimization problems. Several special members of the family are analyzed and the (sufficient) descent condition is established for them (in the vector sense). Under mild conditions, a general convergence result for the CG methods with specific parameters is presented, which, in particular, covers the global convergence of the aforementioned members. Furthermore, for the purpose of comparison, we then consider the direct extension versions of some Dai-Yuan type methods which are obtained by modifying the DY method of the scalar case. These vector extensions can retrieve the classical parameters in the scalar minimization case and their descent property and global convergence are also studied under mild assumptions. Finally, numerical experiments are given to illustrate the practical behavior of all proposed methods.

关键词： Vector optimization Conjugate gradient method Sufficient descent condition Global convergence line search algorithm

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NONlineAR CONJUGATE GRADIENT METHODS FOR VECTOR OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2018年第3期28卷 2690-2720页

作者： Lucambio Perez, L. R. Prudente, L. F. Univ Fed Goias Inst Matemat & Estat Ave Esperanca S-NCampus Samambaia BR-74690900 Goiania Go Brazil

In this work, we propose nonlinear conjugate gradient methods for finding critical points of vector-valued functions with respect to the partial order induced by a closed, convex, and pointed cone with nonempty interior. No convexity assumption is made on the objectives. The concepts of Wolfe and Zoutendjik conditions are extended for the vector-valued optimization. In particular, we show that there exist intervals of step sizes satisfying the Wolfe-type conditions. The convergence analysis covers the vector extensions of the Fletcher-Reeves, conjugate descent, Dai-Yuan, Polak-Ribiere-Polyak, and Hestenes-Stiefel parameters that retrieve the classical ones in the scalar minimization case. Under inexact line searches and without regular restarts, we prove that the sequences generated by the proposed methods find points that satisfy the first-order necessary condition for Pareto-optimality. Numerical experiments illustrating the practical behavior of the methods are presented.

关键词： vector optimization Pareto-optimality conjugate gradient method unconstrained optimization line search algorithm Wolfe conditions

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On the extension of the Hager-Zhang conjugate gradient method for vector optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2020年第3期76卷 889-916页

作者： Goncalves, M. L. N. Prudente, L. F. Univ Fed Goias Inst Matemat & Estat Ave Esperanca S-NCampus Samambaia BR-74690900 Goiania Go Brazil

The extension of the Hager-Zhang (HZ) nonlinear conjugate gradient method for vector optimization is discussed in the present research. In the scalar minimization case, this method generates descent directions whenever, for example, the line search satisfies the standard Wolfe conditions. We first show that, in general, the direct extension of the HZ method for vector optimization does not yield descent (in the vector sense) even when an exact line search is employed. By using a sufficiently accurate line search, we then propose a self-adjusting HZ method which possesses the descent property. The proposed HZ method with suitable parameters reduces to the classical one in the scalar minimization case. Global convergence of the new scheme is proved without regular restarts and any convex assumption. Finally, numerical experiments illustrating the practical behavior of the approach are presented, and comparisons with the Hestenes-Stiefel conjugate gradient and the steepest descent methods are discussed.

关键词： Vector optimization Pareto-optimality Conjugate gradient method Hager-Zhang conjugate gradient method Unconstrained optimization line search algorithm

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Multiple Shooting-Local linearization method for the identification of dynamical systems

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COMMUNICATIONS IN NONlineAR SCIENCE AND NUMERICAL SIMULATION 2016年 37卷 292-304页

作者： Carbonell, F. Iturria-Medina, Y. Jimenez, J. C. Biospective Inc Montreal PQ Canada McGill Univ Montreal Neurol Inst Montreal PQ Canada Inst Cybernet Math & Phys Havana Cuba

The combination of the multiple shooting strategy with the generalized Gauss-Newton algorithm turns out in a recognized method for estimating parameters in ordinary differential equations (ODEs) from noisy discrete observations. A key issue for an efficient implementation of this method is the accurate integration of the ODE and the evaluation of the derivatives involved in the optimization algorithm. In this paper, we study the feasibility of the Local linearization (LL) approach for the simultaneous numerical integration of the ODE and the evaluation of such derivatives. This integration approach results in a stable method for the accurate approximation of the derivatives with no more computational cost than that involved in the integration of the ODE. The numerical simulations show that the proposed Multiple Shooting-Local linearization method recovers the true parameters value under different scenarios of noisy data. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Multiple Shooting Local linear approximation Nonlinear equations Parameter estimation Chaotic dynamics Generalized Gauss-Newton line search algorithm

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Quantitative analysis of upset texture by crystal plasticity: finite element analysis

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INTERNATIONAL JOURNAL OF MATERIALS & PRODUCT TECHNOLOGY 2008年第4期33卷 361-375页

作者： Lee, M. G. Korea Inst Mat Sci Ferrous Alloys Res Grp Chang Won 641010 Geongnam South Korea

The effect of friction and strain state in upset deformation on deformed texture was investigated by crystal plasticity finite element method. It was observed that around 60% among random initial texture evolves to < 111 > within 10 degrees of the upset axis(ND) by 80% upset near the quasi-uniaxial regions, while smearing of this texture components was predicted near the shear dominant regions. Texture intensity plot was introduced to quantitatively investigate the distributions of deformed textures. In addition, a simple Schmid analysis was also performed to explain the origin of strong ND parallel to < 111 > and ND parallel to < 100 > textures from randomly distributed grains.

关键词： finite element crystal plasticity line search algorithm texture intensity

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