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A Hybrid Conjugate Gradient Algorithm for nonconvex functions and Its Applications in Image Restoration Problems

Journal of the Operations Research Society of China 2023年第4期11卷 759-781页

作者： Gong-Lin Yuan Ying-Jie Zhou Meng-Xiang Zhang College of Mathematics and Information Science Center for Applied Mathematics of GuangxiGuangxi UniversityNanning530004China

It is prominent that conjugate gradient method is a high-efficient solution way for large-scale optimization ***,most of the conjugate gradient methods do not have sufficient descent *** this paper,without any line search,the presented method can generate sufficient descent directions and trust region *** use some suitable conditions,the global convergence of the method is established with Armijo line ***,we study the proposed method for solving nonsmooth problems and establish its global *** experiments show that the presented method can be applied to solve smooth and nonsmooth unconstrained problems,image restoration problems and Muskingum model successfully.

关键词： Conjugate gradient Smooth and nonsmooth problems nonconvex functions Global convergence

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GLOBAL CONVERGENCE OF A MODIFIED BROYDEN FAMILY METHOD FOR nonconvex functions

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JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2022年第6期18卷 4393-4407页

作者： Yuan, Gonglin Wang, Zhan Li, Pengyuan Guangxi Univ Coll Math & Informat Sci Ctr Appl Math Guangxi Nanning 530004 Guangxi Peoples R China

The Broyden family method is one of the most effective methods for solving unconstrained optimization problems. However, the study of the global convergence of the Broyden family method is not sufficient. In this paper, a new Broyden family method is proposed based on the BFGS formula of Yuan and Wei (Comput. Optim. Appl. 47: 237-255, 2010). The following approaches are used in the designed algorithm: (1) a modified Broyden family formula is given, (2) every matrix sequence {B-k} generated by the new algorithm possesses positive-definiteness, and (3) the global convergence of the new presented Broyden family algorithm with the Y-W-L inexact line search is obtained for general functions. Numerical performance shows that the modified Broyden family method is competitive with the classical Broyden family method.

关键词： Broyden family method unconstrained optimization global convergence nonconvex functions

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Adaptive three-term PRP algorithms without gradient Lipschitz continuity condition for nonconvex functions

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NUMERICAL ALGORITHMS 2022年第1期91卷 145-160页

作者： Yuan, Gonglin Yang, Heshu Zhang, Mengxiang Guangxi Univ Coll Math & Informat Sci Ctr Appl Math Guangxi Nanning 530004 Guangxi Peoples R China

At present, many conjugate gradient methods with global convergence have been proposed in unconstrained optimization, such as MPRP algorithm proposed by Zhang et al. (IMA J. Numer. Anal. 26(4):629-640, 2006). Unfortunately, almost all of these methods require gradient Lipschitz continuity condition. As far as we know, how do the current conjugate gradient methods deal with gradient non-Lipschitz continuity problems is basically blank. For gradient non-Lipschitz continuity problems, Algorithm 1 and Algorithm 2 are proposed in this paper based on MPRP algorithm. The proposed algorithms have the following characteristics: (i) Algorithm 1 retains sufficient descent property independent of line search technology in MPRP algorithm;(ii) for nonconvex and gradient non-Lipschitz continuous functions, the global convergence of Algorithm 1 is obtained in combination with the trust region property and the weak Wolfe-Powell line search technique;(iii) based on Algorithm 1, Algorithm 2 is further improved which global convergence can be obtained independently of line search technique;(iv) according to numerical experiments, the proposed algorithms perform competitively with other similar algorithms.

关键词： Conjugate gradient nonconvex functions Sufficient descent property Global convergence Gradient Lipschitz continuity condition

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The global convergence of the BFGS method with a modified WWP line search for nonconvex functions

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NUMERICAL ALGORITHMS 2022年第1期91卷 353-365页

作者： Yuan, Gonglin Li, Pengyuan Lu, Junyu Guangxi Univ Coll Math & Informat Sci Ctr Appl Math Guangxi Nanning Guangxi Peoples R China

The BFGS method, which has great numerical stability, is one of the quasi-Newton line search methods. However, the global convergence of the BFGS method with a Wolfe line search is still an open problem for general functions. Recently, Yuan et al. (Appl. Math. Mode. 47:811-825, 2017) presented a modified weak Wolfe-Powell (WWP) line search that globally converges for general functions;however, they only partially solved this open problem. In this paper, a further modified WWP line search is proposed, and the global convergence of the BFGS method is established under suitable conditions for general functions. The numerical results show that the new line search produces more interesting results than the normal line search. Moreover, a fact engineering problem is studied with the Muskingum model to show the performance of the presented algorithm.

关键词： BFGS method Line search Global convergence nonconvex functions

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Family weak conjugate gradient algorithms and their convergence analysis for nonconvex functions

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NUMERICAL ALGORITHMS 2020年第3期84卷 935-956页

作者： Yuan, Gonglin Wang, Xiaoliang Sheng, Zhou Guangxi Univ Coll Math & Informat Sci Nanning 530004 Guangxi Peoples R China Dalian Univ Technol Sch Math Sci Dalian 116024 Liaoning Peoples R China Nanjing Univ Aeronaut & Astronaut Dept Math Nanjing 211106 Jiangsu Peoples R China

It is well-known that conjugate gradient algorithms are widely applied in many practical fields, for instance, engineering problems and finance models, as they are straightforward and characterized by a simple structure and low storage. However, challenging problems remain, such as the convergence of the PRP algorithms for nonconvexity under an inexact line search, obtaining a sufficient descent for all conjugate gradient methods, and other theory properties regarding global convergence and the trust region feature for nonconvex functions. This paper studies family conjugate gradient formulas based on the six classic formulas, PRP, HS, CD, FR, LS, and DY, where the family conjugate gradient algorithms have better theory properties than those of the formulas by themselves. Furthermore, this technique of the presented conjugate gradient formulas can be extended to any two-term conjugate gradient formula. This paper designs family conjugate gradient algorithms for nonconvex functions, which have the following features without other conditions: (i) the sufficient descent property holds, (ii) the trust region feature is true, and (iii) the global convergence holds under normal assumptions. Numerical results show that the given conjugate gradient algorithms are competitive with those of normal methods.

关键词： nonconvex functions Sufficient descent Trust region Inexact line search Global convergence

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The global convergence of the Polak-Ribiere-Polyak conjugate gradient algorithm under inexact line search for nonconvex functions

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2019年第0期362卷 262-275页

作者： Yuan, Gonglin Wei, Zengxin Yang, Yuning Guangxi Univ Coll Math & Informat Sci Nanning Guangxi Peoples R China

Powell (1984) and Dai (2003) constructed respectively a counterexample to show that the Polak-Ribiere-Polyak (PRP) conjugate gradient algorithm fails to globally converge for nonconvex functions even when the exact line search technique is used, which implies similar failure of the weak Wolfe-Powell (WWP) inexact line search technique. Does another inexact line search technique exist that can ensure global convergence for nonconvex functions? This paper gives a positive answer to this question and proposes a modified WWP (MWWP) inexact line search technique. An algorithm is presented that uses the MWWP inexact line search technique;the next point chi(k+1) generated by the PRP formula is accepted if a positive condition holds, and otherwise, chi(k+1) is defined by a technique for projection onto a parabolic surface. The proposed PRP conjugate gradient algorithm is shown to possess global convergence under inexact line search for nonconvex functions. Numerical performance of the proposed algorithm is shown to be competitive with those of similar algorithms. (C) 2018 Elsevier B.V. All rights reserved.

关键词： PRP method Global convergence Inexact line search nonconvex functions

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Proximal point method for a special class of nonconvex functions on Hadamard manifolds

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OPTIMIZATION 2015年第2期64卷 289-319页

作者： Bento, G. C. Ferreira, O. P. Oliveira, P. R. Univ Fed Goias IME BR-74001970 Goiania Go Brazil Univ Fed Rio de Janeiro COPPE Sistemas BR-21945970 Rio De Janeiro RJ Brazil

In this article, we present the proximal point method for finding minima of a special class of nonconvex function on a Hadamard manifold. The well definedness of the sequence generated by the proximal point method is established. Moreover, it is proved that each accumulation point of this sequence satisfies the necessary optimality conditions and, under additional assumptions, its convergence for a minima is obtained.

关键词： Hadamard manifolds nonconvex functions proximal point method 49M30 90C26

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Two-timescale projection neural networks in collaborative neurodynamic approaches to global optimization and distributed optimization

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NEURAL NETWORKS 2024年 169卷 83-91页

作者： Huang, Banghua Liu, Yang Jiang, Yun-Liang Wang, Jun Zhejiang Normal Univ Sch Math Sci Jinhua 321004 Zhejiang Peoples R China Zhejiang Normal Univ Key Lab Intelligent Educ Technol & Applicat Zhejia Jinhua 321004 Zhejiang Peoples R China Zhejiang Normal Univ Sch Comp Sci & Technol Jinhua 321004 Zhejiang Peoples R China Huzhou Univ Sch Informat Engn Huzhou 313000 Zhejiang Peoples R China City Univ Hong Kong Dept Comp Sci Kowloon Hong Kong Peoples R China City Univ Hong Kong Sch Data Sci Kowloon Hong Kong Peoples R China

In this paper, we propose a two-timescale projection neural network (PNN) for solving optimization problems with nonconvex functions. We prove the convergence of the PNN with sufficiently different timescales to a local optimal solution. We develop a collaborative neurodynamic approach with multiple such PNNs to search for global optimal solutions. In addition, we develop a collaborative neurodynamic approach with multiple PNNs connected via a directed graph for distributed global optimization. We elaborate on four numerical examples to illustrate the characteristics of the approaches.

关键词： Collaborative neurodynamic optimization Global optimization Distributed optimization nonconvex functions Two-timescale systems

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A Collaborative Neurodynamic Optimization Approach to Distributed Chiller Loading

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IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2024年第8期35卷 10950-10960页

作者： Chen, Zhongying Wang, Jun Han, Qing-Long City Univ Hong Kong Dept Comp Sci Hong Kong Peoples R China City Univ Hong Kong Dept Comp Sci Hong Kong Peoples R China City Univ Hong Kong Schoolof Data Sci Hong Kong Peoples R China Swinburne Univ Technol Sch Sci Comp & Engn Technol Melbourne Vic 3122 Australia

In this article, we present a collaborative neurodynamic optimization approach to distributed chiller loading in the presence of nonconvex power consumption functions and binary variables associated with cardinality constraints. We formulate a cardinality-constrained distributed optimization problem with nonconvex objective functions and discrete feasible regions, based on an augmented Lagrangian function. To overcome the difficulty caused by the nonconvexity in the formulated distributed optimization problem, we develop a collaborative neurodynamic optimization method based on multiple coupled recurrent neural networks reinitialized repeatedly using a meta-heuristic rule. We elaborate on experimental results based on two multi-chiller systems with the parameters from the chiller manufacturers to demonstrate the efficacy of the proposed approach in comparison to several baselines.

关键词： Optimization Loading Neurodynamics Linear programming HVAC Collaboration Upper bound Collaborative neurodynamic optimization distributed optimization HVAC systems nonconvex functions optimal chiller loading

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An adaptive projection BFGS method for nonconvex unconstrained optimization problems

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NUMERICAL ALGORITHMS 2024年第4期95卷 1747-1767页

作者： Yuan, Gonglin Zhao, Xiong Liu, Kejun Chen, Xiaoxuan Guangxi Univ Ctr Appl Math Guangxi Sch Math & Informat Sci Nanning 530004 Guangxi Peoples R China

The BFGS method is a common and effective method for solving unconstrained optimization problems in quasi-Newton algorithm. However, many scholars have proven that the algorithm may fail in some cases for nonconvex problems under Wolfe conditions. In this paper, an adaptive projection BFGS algorithm is proposed naturally which can solve nonconvex problems, and the following properties are shown in this algorithm: ? The generation of the step size aj satisfies the popular Wolfe conditions;? a specific condition is proposed which has sufficient descent property, and if the current point satisfies this condition, the ordinary BFGS iteration process proceeds as usual;? otherwise, the next iteration point x(j+1) is generated by the proposed adaptive projection method. This algorithm is globally convergent for nonconvex problems under the weak-Wolfe-Powell (WWP) conditions and has a superlinear convergence rate, which can be regarded as an extension of projection BFGS method proposed by Yuan et al. (J. Comput. Appl. Math. 327:274-294, 2018). Furthermore, the final numerical results and the application of the algorithm to the Muskingum model demonstrate the feasibility and competitiveness of the algorithm.

关键词： BFGS method nonconvex functions Adaptive projection technique Global convergence Muskingum model

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