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A NON-MONOTONE smoothing newton algorithm FOR SOLVING THE SYSTEM OF GENERALIZED ABSOLUTE VALUE EQUATIONS

Journal of Computational Mathematics 2025年第2期43卷 438-460页

作者： Cairong Chen Dongmei Yu Deren Han Changfeng Ma School of Mathematics and Statistics Key Laboratory of Analytical Mathematics and Applications(Ministry of Education)Fujian Provincial Key Laboratory of Statistics and Artificial IntelligenceFujian Normal UniversityFuzhou 350117P.R.China Institute for Optimization and Decision Analytics College of ScienceLiaoning Technical UniversityFuxin 123000P.R.China LMIB of the Ministry of Education School of Mathematical SciencesBeihang UniversityBeijing 100191P.R.China

The system of generalized absolute value equations(GAVE)has attracted more and more attention in the optimization *** this paper,by introducing a smoothing function,we develop a smoothing newton algorithm with non-monotone line search to solve the *** show that the non-monotone algorithm is globally and locally quadratically convergent under a weaker assumption than those given in most existing algorithms for solving the *** results are given to demonstrate the viability and efficiency of the approach.

关键词： Generalized absolute value equations smoothing function smoothing newton algorithm Non-monotone line search Global and local quadratic convergence

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The solvability of weighted complementarity problems and a smoothing newton algorithm under the local error bound

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OPTIMIZATION 2025年第3期74卷 811-841页

作者： Tang, Jingyong Zhou, Jinchuan Xinyang Normal Univ Coll Math & Stat Xinyang Peoples R China Shandong Univ Technol Coll Math & Stat Zibo Peoples R China

The weighted Complementarity Problem (wCP) can be used for modelling a larger class of problems from science and engineering. In this paper, we first extend the solvability of the Complementarity Problems (CP) obtained by Yoshise [SIAM J. Optim. 17, 1129-1153 (2006)] to the wCP. Especially, we prove that the solution set of the wCP is bounded. We then propose a smoothing newton algorithm to solve the wCP and prove that it is globally convergent. Moreover, we establish the local quadratic convergence of the proposed algorithm under the local error bound condition which is weaker than the nonsingularity condition used in previous smoothing newton-type algorithms. Further, we apply the proposed algorithm to solve the weighted horizontal linear complementarity problem and report some numerical results.

关键词： Weighted complementarity problems solvability smoothing newton algorithm local error bound quadratic convergence

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A Nonmonotone smoothing newton algorithm for Weighted Complementarity Problem

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2021年第3期189卷 679-715页

作者： Tang, Jingyong Zhang, Hongchao Xinyang Normal Univ Sch Math & Stat Xinyang 464000 Peoples R China Louisiana State Univ Dept Math 303 Lockett Hall Baton Rouge LA 70803 USA

The weighted complementarity problem (denoted by WCP) significantly extends the general complementarity problem and can be used for modeling a larger class of problems from science and engineering. In this paper, by introducing a one-parametric class of smoothing functions which includes the weight vector, we propose a smoothing newton algorithm with nonmonotone line search to solve WCP. We show that any accumulation point of the iterates generated by this algorithm, if exists, is a solution of the considered WCP. Moreover, when the solution set of WCP is nonempty, under assumptions weaker than the Jacobian nonsingularity assumption, we prove that the iteration sequence generated by our algorithm is bounded and converges to one solution of WCP with local superlinear or quadratic convergence rate. Promising numerical results are also reported.

关键词： smoothing newton algorithm Jacobian nonsingularity Superlinear quadratic convergence Weighted complementarity problem Symmetric cone

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A cosh-based smoothing newton algorithm for the real-time pricing problem in smart grid

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INTERNATIONAL JOURNAL OF ELECTRICAL POWER & ENERGY SYSTEMS 2022年 135卷 107296-107296页

作者： Li, Yuanyuan Li, Junxiang Yu, Zhensheng Dong, Jingxin Zhou, Tingting Univ Shanghai Sci & Technol Business Sch Shanghai 200093 Peoples R China Univ Shanghai Sci & Technol Coll Sci Shanghai 200093 Peoples R China Newcastle Univ Business Sch Newcastle Upon Tyne Tyne & Wear England

In a smart grid system, different types of electricity users such as residential, commercial and industrial users have different utility functions. This paper proposes a nonlinear constrained optimization model for real-time pricing that maximizes the total utilities of all the different user groups. A Karush-Kuhn-Tucker equation system is employed to solve the model. However, it is a challenging task to solve the system due to the nonlinear complementary condition. To tackle the challenge, a cosh-based smoothing approximation function is proposed to substitute the nonlinear complementary condition. Subsequently, the smoothing newton algorithm is developed to solve the new equation system. The global convergence and local quadratic convergence of the algorithm are proved. Numerical experiments are performed to test the smoothing method and compare the solutions of real-time pricing, fixed pricing and time-of-use pricing strategies applied in the smart grid system. The results show that the real-time pricing mechanism is the most suitable in saving energy and reducing peaks and troughs in energy consumption. This also indicates that it is effective to use the smoothing newton algorithm to solve the problem of real-time electricity pricing for smart grid. We obtain that the smoothing approximation function is effective for the supply-demand balance constraints in smart grid.

关键词： Smart grid smoothing function Real-time pricing smoothing newton algorithm Global convergence

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smoothing newton algorithm for the second-order cone programming with a nonmonotone line search

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OPTIMIZATION LETTERS 2014年第5期8卷 1753-1771页

作者： Tang, Jingyong Dong, Li Fang, Liang Zhou, Jinchuan Xinyang Normal Univ Coll Math & Informat Sci Xinyang 464000 Peoples R China Taishan Univ Coll Math & Syst Sci Tai An 271021 Shandong Peoples R China Shandong Univ Technol Sch Sci Dept Math Zibo 255049 Peoples R China

The smoothing-type algorithms, which are in general designed based on some monotone line search, have been successfully applied to solve the second-order cone programming (denoted by SOCP). In this paper, we propose a nonmonotone smoothing newton algorithm for solving the SOCP. Under suitable assumptions, we show that the proposed algorithm is globally and locally quadratically convergent. To compare with the existing smoothing-type algorithms for the SOCP, our algorithm has the following special properties: (i) it is based on a new smoothing function of the vector-valued natural residual function;(ii) it uses a nonmonotone line search scheme which contains the usual monotone line search as a special case. Preliminary numerical results demonstrate that the smoothing-type algorithm using the nonmonotone line search is promising for solving the SOCP.

关键词： Second-order cone programming smoothing newton algorithm Nonmonotone line search Convergence

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A smoothing newton algorithm for a Class of Non-monotonic Symmetric Cone Linear Complementarity Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第2期161卷 446-464页

作者： Lu, Nan Huang, Zheng-Hai Xidian Univ Dept Math Xian 710071 Peoples R China Tianjin Univ Sch Sci Dept Math Tianjin 300072 Peoples R China Tianjin Univ Ctr Appl Math Tianjin 300072 Peoples R China

Recently, the study of symmetric cone complementarity problems has been a hot topic in the literature. Many numerical methods have been proposed for solving such a class of problems. Among them, the problems concerned are generally monotonic. In this paper, we consider symmetric cone linear complementarity problems with a class of non-monotonic transformations. A smoothing newton algorithm is extended to solve this class of non-monotonic symmetric cone linear complementarity problems;and the algorithm is proved to be well-defined. In particular, we show that the algorithm is globally and locally quadratically convergent under mild assumptions. The preliminary numerical results are also reported.

关键词： Symmetric cone complementarity problem Euclidean Jordan algebra smoothing newton algorithm Global convergence Local quadratic convergence

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A Modified smoothing newton Method for Solving Weighted Complementarity Problems with a Nonmonotone Line Search

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ENGINEERING LETTERS 2023年第4期31卷 1674页

作者： Liu, Xiangjing Zhang, Jianke Chen, Junfeng Xian Technol Univ Sch Sci Xian 710021 Peoples R China Xian Univ Post & Telecommun Sch Sci Xian 710121 Peoples R China Xianyang Normal Univ Sch Math & Stat Xianyang 712000 Peoples R China

we concentrate on the general form of the weighted complementarity problem, which serves as a generalization of the nonlinear complementarity problem and finds extensive applications in various fields, including economics, sciences, engineering, atmospheric chemistry, and multibody dynamics. We introduce a novel Fischer-Burmeister-based one-parameter smoothing complementarity function. The WCP is then reformulated as a smoothing system of equations, and a new smoothing newton method is devised to solve the problem efficiently on the new one-parameter smoothing complementarity function. To ensure global convergence, we introduce a new line search rule. The new method exhibits both global and local quadratic convergence properties under appropriate conditions, as demonstrated through several numerical experiments that confirm its effectiveness and stability.

关键词： Index Terms-weighted complementarity problem nonmono-tone line search smoothing newton algorithm convergence analysis

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A note on quadratic convergence of a smoothing newton algorithm for the LCP

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OPTIMIZATION LETTERS 2013年第3期7卷 519-531页

作者： Ni, Tie Hu, Sheng-Long Liaoning Tech Univ Coll Business Adm Huludao 125105 Peoples R China Tianjin Univ Sch Sci Dept Math Tianjin 300072 Peoples R China

The linear complementarity problem (LCP) is to find such that (x, s) a parts per thousand yen 0, s = Mx + q, x (T) s = 0 with and . The smoothing newton algorithm is one of the most efficient methods for solving the LCP. To the best of our knowledge, the best local convergence results of the smoothing newton algorithm for the LCP up to now were obtained by Huang et al. (Math Program 99:423-441, 2004). In this note, by using a revised Chen-Harker-Kanzow-Smale smoothing function, we propose a variation of Huang-Qi-Sun's algorithm and show that the algorithm possesses better local convergence properties than those given in Huang et al. (Math Program 99:423-441, 2004).

关键词： Linear complementarity problem smoothing newton algorithm Quadratic convergence

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A new smoothing-type algorithm for nonlinear weighted complementarity problem

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2020年第1-2期64卷 215-226页

作者： Liu, Ziyu Tang, Jingyong Xinyang Normal Univ Sch Math & Stat Xinyang 464000 Peoples R China

In this paper we study the nonlinear weighted complementarity problem (denoted by NWCP). We first introduce a smoothing function which can be used to reformulate the NWCP as a system of smooth nonlinear equations. Then we propose a new smoothing-type algorithm to solve the NWCP which adopts a nonmonotone line search scheme. In each iteration, our algorithm solves one linear system of equations and performs one line search. Under suitable assumptions, we prove that the proposed algorithm is globally and locally quadratically convergent. Some numerical results are reported.

关键词： Nonlinear weighted complementarity problem smoothing newton algorithm Nonmonotone line search Quadratical convergence

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New smoothing function for solving absolute value equations

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ASIAN-EUROPEAN JOURNAL OF MATHEMATICS 2023年第12期16卷

作者： Chalekh, Randa Djeffal, El Amir Univ Batna 2 Math Dept LEDPA Lab Batna 05000 Algeria

In this paper, we use a smoothing-type algorithm in this paper to solve the AVE, which stands for the absolute value equation Ax-|x|=b , where A is an arbitrary nxn real matrix and b is an element of R-n. We reformulate AVE as a system of smooth equations and propose two new smoothing functions. We prove that the algorithm is well-defined when the singular value of A exceeds one, and under the same assumption, the algorithm is convergent. We show the algorithm's effectiveness with these two functions and compare it with some previously known functions.

关键词： Absolute value equations smoothing newton algorithm smoothing function convergence

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