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Probability-directed random search algorithm for unconstrained optimization problem

APPLIED SOFT COMPUTING 2018年 71卷 165-182页

作者： Al-Muhammed, Muhammed Jassem Abu Zitar, Raed Amer Univ Madaba Fac Informat Technol Madaba Jordan Ajman Univ Ajman Coll Informat Technol Ajman U Arab Emirates

Devising ways for handling problem optimization is an important yet a challenging task. The aims are always for methods that can effectively and quickly discover the global optimum for rather complicated mathematical functions that model real-world settings. Typically these functions are too difficult to discover their global optima because they may (1) lack the continuity and differentiability, (2) have multiple local optima, and (3) have complex expressions. In this paper, we address this challenge by offering an algorithm that combines the random search techniques with both an effective mapping and a dynamic adjustment of its search behavior. Our proposed algorithm automatically builds two types of triangular search directives over the unity intervals: principal and marginal. These search directives guide the search within both the effective regions of the search domain that most likely contain the optimum and the marginal regions of the search domain that less likely contain the optimum. During the search the algorithm monitors the intermediate search results and dynamically adjusts the directives' parameters to quickly move the search towards the optimum. Experiments with our prototype implementation showed that our method can effectively find the global optima for rather complicated mathematical functions chosen from well-known benchmarks, and performed better than other algorithms. (C) 2018 Elsevier B.V. All rights reserved.

关键词： Probability-directed optimization unconstrained optimization problem Global optimum Multiple local optima

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Two sufficient descent three-term conjugate gradient methods for unconstrained optimization problems with applications in compressive sensing

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2022年第3期68卷 1787-1816页

作者： Liu, Yufeng Zhu, Zhibin Zhang, Benxin Guilin Univ Elect Technol Sch Elect Engn & Automat Guangxi Key Lab Automat Detecting Technol & Instr Guilin 541004 Peoples R China Guilin Univ Elect Technol Sch Math & Comp Sci Guangxi Coll & Univ Key Lab Data Anal & Computat Guilin 541004 Peoples R China

In this paper, we present two new three-term conjugate gradient methods which can generate sufficient descent directions for the large-scale optimization problems. Note that this property is independent of the line search used. We prove that these three-term conjugate gradient methods are global convergence under the Wolfe line search. Numerical experiments and comparisons demonstrate that the proposed algorithms are efficient approaches for test functions. Moreover, we use the proposed methods to solve the l(1) - alpha l(2) regularization problem of sparse signal decoding in compressed sensing, and the results show that our methods have certain advantages over the existing solvers on such problems.

关键词： unconstrained optimization problem Three-term conjugate gradient method Sufficient descent property Global convergence l(1) - alpha l(2) minimization Compressive sensing

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Two spectral conjugate gradient methods for unconstrained optimization problems

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2022年第6期68卷 4821-4841页

作者： Zhu, Zhibin Long, Ai Wang, Tian Guilin Univ Elect Technol Sch Math & Comp Sci Guangxi Coll & Univ Key Lab Data Anal & Computat Guilin 541004 Peoples R China

Two new spectral conjugate gradient methods (ZL1 method and ZL2 method) for solving unconstrained optimization problems are established. Under the standard Wolfe line search, the search direction generated by the ZL1 method is a descent direction. The search direction of the ZL2 method satisfies descent property independent of the line search. The global convergence of the two new methods can be demonstrated under the standard Wolfe line search. Numerical experiments are presented to show that the two methods are effective.

关键词： unconstrained optimization problem Spectral conjugate gradient method Standard Wolfe line search Global convergence

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An adaptive restarting RMIL conjugate gradient method and its hybrid version: global convergence and iterative complexity analyses

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JOURNAL OF APPLIED MATHEMATICS AND COMPUTING 2025年 1-22页

作者： Ma, Xuejie Wu, Bin Liu, Pengjie He, Bo Guangzhou Huashang Coll Sch Artificial Intelligence Guangzhou 511300 Peoples R China China Univ Min & Technol Sch Math Xuzhou 221116 Peoples R China Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Peoples R China

In this study, we present an adaptive restarting RMIL conjugate gradient (CG) method and its hybrid version for solving unconstrained optimization problems. First, we introduce an effective restarting direction and integrate it into the RMIL CG method. The resulting composite search direction, independent of any line search, possesses the sufficient descent property. Then, by using the hybrid strategy presented in [Applied Mathematical Modelling 39 (2015) 1281-1290], we obtain a hybrid RMIL CG method with a restarting mechanism, whose search direction also has the desired properties. Furthermore, we demonstrate the global convergence of the two proposed methods under conventional assumptions, where the step sizes are determined using the weak Wolfe line search. Additionally, by using the Armijo line search to determine step sizes, we derive the iterative complexity of the obtained methods. Numerical experiments comparing our methods with existing methods confirm their efficiency in addressing unconstrained optimization problems. Finally, we employ the proposed methods for image restoration tasks.

关键词： unconstrained optimization problem RMIL conjugate gradient method Restart mechanism Theoretical property Image restoration

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Adjustment of the scaling parameter of Dai-Kou type conjugate gradient methods with application to motion control

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APPLICATIONS OF MATHEMATICS 2024年第6期69卷 829-845页

作者： Akbari, Mahbube Nezhadhosein, Saeed Heydari, Aghile Payame Noor Univ Dept Math Nakhl StLashkarak Highway Tehran 193954697 Iran

We introduce a new scaling parameter for the Dai-Kou family of conjugate gradient algorithms (2013), which is one of the most numerically efficient methods for unconstrained optimization. The suggested parameter is based on eigenvalue analysis of the search direction matrix and minimizing the measure function defined by Dennis and Wolkowicz (1993). The corresponding search direction of conjugate gradient method has the sufficient descent property and the extended conjugacy condition. The global convergence of the proposed algorithm is given for both uniformly convex and general nonlinear objective functions. Also, numerical experiments on a set of test functions of the CUTER collections and the practical problem of the manipulator of robot movement control show that the proposed method is effective.

关键词： unconstrained optimization problem self-scaling memoryless BFGS conjugate gradient measure function

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MEMORYLESS THREE-TERM RMIL-TYPE CONJUGATE GRADIENT METHOD FOR SOLVING unconstrained optimization AND IMAGE DENOISING problemS

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JOURNAL OF INDUSTRIAL AND MANAGEMENT optimization 2025年第6期21卷 4575-4589页

作者： Ishak, Muhammad Aqiil Iqmal Marjugi, Siti Mahani June, Leong Wah Univ Putra Malaysia Fac Sci Dept Math & Stat Upm Serdang 43400 Selangor Malaysia

. This study introduces a modified version of the Rivaie-MustafaIsmail-Leong (RMIL) Conjugate Gradient Method (CGM), which produces a search direction similar to that of the memoryless BFGS Quasi-Newton approach. When exact line search (ELS) is applied, the method reverts to the original RMIL method. Additionally, for any line search with a constant t is an element of [0,1), the proposed method's direction fulfills the sufficient descent condition (SDC). The Global Convergence (GC) of the method is demonstrated for strongly convex objective functions using the weak Wolfe line search (WWLS). The effectiveness of the proposed methods are evaluated through two approaches. First, numerical experiments are conducted on unconstrained optimization problem (UOP), where the proposed method displays comparable performance to existing CGMs which significantly outperforms under WWLS, achieving faster convergence. Second, the proposed methods are applied to Image Denoising problem (IDP), further validating its practical applications. The results indicate that the proposed method holds strong theoretical and numerical promise for optimization challenges.

关键词： unconstrained optimization problem wolfe line search conjugate gra-dient method memoryless BFGS method RMIL method sufficient descent global convergence image denoising

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A novel method for solving universum twin bounded support vector machine in the primal space

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ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE 2025年第1期93卷 131-150页

作者： Moosaei, Hossein Khosravi, Saeed Bazikar, Fatemeh Hladik, Milan Guarracino, Mario Rosario Univ JE Purkyne Fac Sci Dept Informat Usti Nad Labem Czech Republic Charles Univ Prague Fac Math & Phys Sch Comp Sci Dept Appl Math Prague Czech Republic Prague Univ Econ & Business Prague Czech Republic Univ Bojnord Fac Sci Dept Comp Sci Bojnord Iran Univ Guilan Fac Math Sci Dept Appl Math Rasht Iran Charles Univ Prague Fac Math & Phys Dept Appl Math Prague Czech Republic Univ Econ Dept Econometr Prague Czech Republic Univ Cassino & Southern Lazio Campus Folcara Dept Econ & Law Cassino Italy Natl Res Univ Higher Sch Econ Lab Algorithms & Technol Networks Anal Nizhnii Novgorod Russia

In supervised learning, the Universum, a third class that is not a part of either class in the classification task, has proven to be useful. In this study we propose (NU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak {U} $$\end{document}TBSVM), a Newton-based approach for solving in the primal space the optimization problems related to Twin Bounded Support Vector Machines with Universum data (U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak {U} $$\end{document}TBSVM). In the NU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak {U} $$\end{document}TBSVM, the constrained programming problems of U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak {U} $$\end{document}TBSVM are converted into unconstrained optimization problems, and a generalization of Newton's method for solving the unconstrained problems is introduced. Numerical experiments on synthetic, UCI, and NDC data sets show the ability and effectiveness of the proposed NU\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathfrak {U} $$\end{document}TBSVM. We apply the suggested met

关键词： Twin bounded support vector machine Universum Newton's method unconstrained optimization problem

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A Novel Method for Solving Universum Twin Bounded Support Vector Machine in the Primal Space

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ANNALS OF MATHEMATICS AND ARTIFICIAL INTELLIGENCE 2023年 1-20页

作者： Moosaei, Hossein Khosravi, Saeed Bazikar, Fatemeh Hladik, Milan Guarracino, Mario Rosario Univ JE Purkyne Fac Sci Dept Informat Usti Nad Labem Czech Republic Charles Univ Prague Fac Math & Phys Sch Comp Sci Dept Appl Math Prague Czech Republic Prague Univ Econ & Business Prague Czech Republic Univ Bojnord Fac Sci Dept Comp Sci Bojnord Iran Univ Guilan Fac Math Sci Dept Appl Math Rasht Iran Charles Univ Prague Fac Math & Phys Dept Appl Math Prague Czech Republic Univ Econ Dept Econometr Prague Czech Republic Univ Cassino & Southern Lazio Dept Econ & Law Campus Folcara Cassino Italy Natl Res Univ Higher Sch Econ Lab Algorithms & Technol Networks Anal Moscow Russia

In supervised learning, the Universum, a third class that is not a part of either class in the classification task, has proven to be useful. In this study we propose (NUTBSVM), a Newton based approach for solving in the primal space the optimization problems related to Twin Bounded Support Vector Machines with Universum data (UTBSVM). In the NUTBSVM, the constrained programming problems of UTBSVM are converted into unconstrained optimization problems, and a generalization of Newton's method for solving the unconstrained problems is introduced. Numerical experiments on synthetic, UCI, and NDC data sets show the ability and effectiveness of the proposed NUTBSVM. We apply the suggested method for gender detection from face images, and compare it with other methods.

关键词： Twin bounded support vector machine Universum Newton's method unconstrained optimization problem

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Two modified DY conjugate gradient methods for unconstrained optimization problems

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APPLIED MATHEMATICS AND COMPUTATION 2020年第0期373卷 125004-000页

作者： Zhu, Zhibin Zhang, Dongdong Wang, Shuo Guilin Univ Elect Technol Guangxi Coll & Univ Key Lab Data Anal & Computat Sch Math & Comp Sci Guilin 541004 Peoples R China Guilin Univ Elect Technol Sch Elect Engn & Automat Guilin 541004 Peoples R China

In this paper, we study the unconstrained optimization problems, and the two modified DY conjugate gradient methods (DDY1 method and DDY2 method) are proposed under the DY conjugate gradient method. By using the standard Wolfe line search, we prove the global convergence of the two methods. The search direction of DDY1 method is descent with the standard Wolfe line search. The search direction generated by the DDY2 method is sufficient descent, in which the property does not depend on any line search. Preliminary numerical results show that the two methods are effective. (C) 2019 Elsevier Inc. All rights reserved.

关键词： unconstrained optimization problem Conjugate gradient method Standard Wolfe line search Global convergence Sufficient descent property

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Low-complexity linear massive MIMO detection based on the improved BFGS method

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IET COMMUNICATIONS 2022年第14期16卷 1699-1707页

作者： Li, Lin Hu, Jianhao Univ Elect Sci & Technol China Chengdu Sichuan Peoples R China Qinghai Normal Univ Xining Qinghai Peoples R China

Linear minimum mean square error (MMSE) detection achieves a good trade-off between performance and complexity for massive multiple-input multiple-output (MIMO) systems. To avoid the high-dimensional matrix inversion involved, MMSE detection can be transformed into an unconstrained optimization problem and then solved by efficient numerical algorithms in an iterative way. Three low-complexity Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton methods are proposed to iteratively realize massive MIMO MMSE detection without matrix inversion. The complexity can be reduced from O(K3)$\mathcal {O}(K<^>{3})$ to O(LK2)$\mathcal {O}(LK<^>{2})$, where K and L denote the number of users and iterations, respectively. Leveraging the special properties of massive MIMO, the authors first explore a simplified BFGS method (named S-BFGS) to alleviate the computational burden in the search direction. For lower complexity, BFGS method with the unit step size (named U-BFGS) is presented subsequently. When the base station (BS)-to-user-antenna ratio (BUAR) is large enough, the two proposed BFGS methods can be integrated (named U-S-BFGS) to further reduce complexity. In addition, an efficient initialization strategy is devised to accelerate convergence. Simulation results verify that the proposed detection scheme can achieve near-MMSE performance with a small number of iterations L as low as 2 or 3.

关键词： base station-to-user-antenna ratio MIMO communication matrix inversion least mean squares methods optimisation linear minimum mean square error detection simplified BFGS method computational complexity massive MIMO MMSE detection Newton method high-dimensional matrix inversion low-complexity Broyden-Fletcher-Goldfarb-Shanno quasiNewton methods detection scheme improved BFGS method near-MMSE performance iterative methods U-BFGS convergence of numerical methods maximum likelihood detection unconstrained optimization problem low-complexity linear massive MIMO detection massive multiple-input multiple-output systems

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