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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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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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Spectral conjugate gradient methods for vector optimization problems

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2023年第2期86卷 457-489页

作者： He, Qing-Rui Chen, Chun-Rong Li, Sheng-Jie Chongqing Univ Coll Math & Stat Chongqing 401331 Peoples R China

In this work, we present an extension of the spectral conjugate gradient (SCG) methods for solving unconstrained vector optimization problems, with respect to the partial order induced by a pointed, closed and convex cone with a nonempty interior. We first study the direct extension version of the SCG methods and its global convergence without imposing an explicit restriction on parameters. It shows that the methods may lose their good scalar properties, like yielding descent directions, in the vector setting. By using a truncation technique, we then propose a modified self-adjusting SCG algorithm which is more suitable for various parameters. Global convergence of the new scheme covers the vector extensions of three different spectral parameters and the corresponding Perry, Andrei, and Dai-Kou conjugate parameters (SP, N, and JC schemes, respectively) without regular restarts and any convex assumption. Under inexact line searches, we prove that the sequences generated by the proposed methods find points that satisfy the first-order necessary condition for Pareto-optimality. Finally, numerical experiments illustrating the practical behavior of the methods are presented.

关键词： Vector optimization Pareto optimality Spectral conjugate gradient method Unconstrained optimization line search algorithm

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Electric Vehicle Range Estimation and Optimization 4

Electric Vehicle Range Estimation and Optimization

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4th International Conference on Artificial Intelligence and Signal Processing

作者： Aravind, S. Aryan, B. Adithya, R. Anand, R. Amrita Vishwa Vidyapeetham Dept Elect & Elect Engn Amrita Sch Engn Coimbatore Tamil Nadu India

ISBN: (纸本)9798350350661;9798350350654

Accurate range prediction is crucial for electric vehicle (EV) adoption, mitigating range anxiety and optimizing energy usage. This paper presents a novel approach for EV range prediction using a physics-based formula and optimization algorithms. Unlike existing range planning software, our method focuses solely on prediction, employing Particle Swarm Optimization (PSO) and line search algorithm to optimize trip-specific variables within a range, providing best-case and worst-case range scenarios. This paper details the methodology, underlying formula, and future research directions.

关键词： Electric Vehicle Range Prediction Optimization Particle Swarm Optimization line search algorithm

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Sufficient Descent Riemannian Conjugate Gradient Methods

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2021年第1期190卷 130-150页

作者： Sakai, Hiroyuki Iiduka, Hideaki Meiji Univ Dept Comp Sci Tama Ku 1-1-1 Higashimita Kawasaki Kanagawa 2148571 Japan

This paper considers sufficient descent Riemannian conjugate gradient methods with line search algorithms. We propose two kinds of sufficient descent nonlinear conjugate gradient method and prove that these methods satisfy the sufficient descent condition on Riemannian manifolds. One is a hybrid method combining a Fletcher-Reeves-type method with a Polak-Ribiere-Polyak-type method, and the other is a Hager-Zhang-type method, both of which are generalizations of those used in Euclidean space. Moreover, we prove that the hybrid method has a global convergence property under the strong Wolfe conditions and the Hager-Zhang-type method has the sufficient descent property regardless of whether a line search is used or not. Further, we review two kinds of line search algorithm on Riemannian manifolds and numerically compare our generalized methods by solving several Riemannian optimization problems. The results show that the performance of the proposed hybrid methods greatly depends on the type of line search used. Meanwhile, the Hager-Zhang-type method has the fast convergence property regardless of the type of line search used.

关键词： Riemannian conjugate gradient method Sufficient descent condition Strong Wolfe conditions line search algorithm

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Evaluating the Performance of Various ACOPF Formulations Using Nonlinear Interior-Point Method

Evaluating the Performance of Various ACOPF Formulations Usi...

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IEEE International Smart Cities Conference (ISC2)

作者： Sadat, Sayed Abdullah Univ Utah Dept Elect & Comp Engn Salt Lake City UT 84112 USA

ISBN: (纸本)9781665449199

Alternating current optimal power flow (ACOPF) problem is a non-convex and a nonlinear optimization problem. Similar to most nonlinear optimization problems, ACOPF is an NP-hard problem. On the other hand, Utilities and independent service operators (ISO) require the problem to be solved in almost real-time. The real-world networks are often large in size and developing an efficient and tractable algorithm is critical to many decision-making processes in electricity markets. Interior-point methods (IPMs) for nonlinear programming are considered one of the most powerful algorithms for solving large-scale nonlinear optimization problems. However, the performance of these algorithms is significantly impacted by the optimization structure of the problem. Thus, the choice of the formulation is as important as choosing the algorithm for solving an ACOPF problem. Different ACOPF formulations are evaluated in this paper for computational viability and best performance using the interior-point line search (IPLS) algorithm. Different optimization structures are used in these formulations to model the ACOPF problem representing a range of varying sparsity. The numerical experiments suggest that the least sparse ACOPF formulation with polar voltages yields the best computational results. A wide range of test cases, ranging from 500-bus systems to 9591-bus systems, are used to verify the test results.

关键词： ACOPF interior-point methods optimal power flow line search algorithm sparsity power system economics ACOPF formulations

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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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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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