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Eigenvalues versus singular values study in conjugate gradient algorithms for large-scale unconstrained optimization

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OPTIMIZATION METHODS & SOFTWARE 2017年第3期32卷 534-551页

作者： Andrei, Neculai Res Inst Informat Ctr Adv Modeling & Optimizat 8-10 Averescu Ave Bucharest 1 Romania

Two different approaches based on eigenvalues and singular values of the matrix representing the search direction in conjugate gradient algorithms are considered. Using a special approximation of the inverse Hessian of the objective function, which depends by a positive parameter, we get the search direction which satisfies both the sufficient descent condition and the Dai-Liao's conjugacy condition. In the first approach the parameter in the search direction is determined by clustering the eigenvalues of the matrix defining it. The second approach uses the minimizing the condition number of the matrix representing the search direction. In this case the obtained conjugate gradient algorithm is exactly the three-term conjugate gradient algorithm proposed by Zhang, Zhou and Li. The global convergence of the algorithms is proved for uniformly convex functions. Intensive numerical experiments, using 800 unconstrained optimization test problems, prove that both these approaches have similar numerical performances. We prove that both algorithms are significantly more efficient and more robust than CG-DESCENT algorithm by Hager and Zhang. By solving five applications from the MINPACK-2 test problem collection, with 106 variables, we show that the suggested conjugate gradient algorithms are top performer versus CG-DESCENT.

关键词： unconstrained optimization conjugate gradient algorithms eigenvalues singular values Wolfe conditions convergence sufficient descent condition conjugacy condition 49M07 49M10 90C06 65K

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Accelerated adaptive Perry conjugate gradient algorithms based on the self-scaling memoryless BFGS update

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 325卷 149-164页

作者： Andrei, Neculai Ctr Adv Modeling & Optimizat Res Inst Informat 8-10 Averescu Ave Bucharest 1 Romania Acad Romanian Scientists Splaiul Independentei 54Sect 5 Bucharest Romania

An accelerated adaptive class of nonlinear conjugate gradient algorithms is suggested. The search direction in these algorithms is given by symmetrization of the scaled Perry conjugate gradient direction (Perry, 1978), which depends on a positive parameter. The value of this parameter is determined by minimizing the distance between the symmetrical scaled Perry conjugate gradient search direction matrix and the self-scaling memoryless BFGS update by Oren in the Frobenius norm. Two variants of the parameter in the search direction are presented as those given by: Oren and Luenberger (1973/74) and Oren and Spedicato (1976). The corresponding algorithm, ACGSSV, is equipped with a very well known acceleration scheme of conjugate gradient algorithms. The global convergence of the algorithm is given both for uniformly convex and general nonlinear functions under the exact or the Wolfe line search. Using a set of 800 unconstrained optimization test problems, of different structure and complexity, we prove that selection of the scaling parameter in self-scaling memoryless BFGS update leads to algorithms which substantially outperform the CG-DESCENT, SCALCG, and CONMIN conjugate gradient algorithms, being more efficient and more robust. However, the conjugate gradient algorithm ADCG based on clustering the eigenvalues of the iteration matrix defined by the search direction is more efficient and slightly more robust than our ACGSSV algorithm. By solving five applications from the MINPACK-2 test problem collection with 10(6) variables, we show that the adaptive Perry conjugate gradient algorithms based on the self-scaling memoryless BFGS update, endowed with the acceleration scheme, is top performer versus CG_DESCENT. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Unconstrained optimization conjugate gradient algorithms Self-scaling memoryless BFGS update Sufficient descent condition Conjugacy condition Frobenius norm

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Performance Analysis of conjugate gradient algorithms Applied to the Neuro-Fuzzy Feedback Linearization-Based Adaptive Control Paradigm for Multiple HVDC Links in AC/DC Power System

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ENERGIES 2017年第6期10卷 819-819页

作者： Ahmad, Saghir Khan, Laiq COMSATS Inst Informat Technol Dept Elect Engn Abbottabad 22060 Pakistan

The existing literature predominantly concentrates on the utilization of the gradient descent algorithm for control systems' design in power systems for stability enhancement. In this paper, various flavors of the conjugate gradient (CG) algorithm have been employed to design the online neuro-fuzzy linearization-based adaptive control strategy for Line Commutated Converters' (LCC) High Voltage Direct Current (HVDC) links embedded in a multi-machine test power system. The conjugate gradient algorithms are evaluated based on the damping of electro-mechanical oscillatory modes using MATLAB/Simulink. The results validate that all of the conjugate gradient algorithms have outperformed the gradient descent optimization scheme and other conventional and non-conventional control schemes.

关键词： low-frequency oscillations HVDC system adaptive feedback linearization control adaptive neuro-fuzzy inference system conjugate gradient algorithms

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conjugate gradient-TYPE algorithms FOR A FINITE-ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1992年第1期39卷 23-37页

作者： HEUSSER, C UNIV ZURICH INST ANGEW MATHCH-8001 ZURICHSWITZERLAND

The discretization of the Stokes equations with the mini-element yields a linear system of equations whose system matrix is symmetric and indefinite. It has two symmetric blocks, one of which is positive definite, the other negative definite. A change of sign in the second block destroys the symmetry but the resulting matrix is coercive. We discuss different conjugate gradient-like algorithms for these two systems, and compare the storage and work requirements. The numerical results of several test examples are given.

关键词： STOKES EQUATIONS MINIELEMENT conjugate gradient algorithms conjugate RESIDUAL BIconjugate gradient algorithms

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GLOBAL CONVERGENCE RESULT FOR conjugate-gradient METHODS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1991年第2期71卷 399-405页

作者： HU, YF STOREY, C 1. Department of Mathematical Sciences Loughborough University of Technology Loughborough Leicestershire England

conjugate gradient optimization algorithms depend on the search directions, s(1) = -g(1), s(k + 1) = -g(k + 1) + beta(k)s(k), k greater-than-or-equal-to 1, with different methods arising from different choices for the scalar beta(k). In this note, conditions are given on beta(k) to ensure global convergence of the resulting algorithms.

关键词： conjugate gradient algorithms GLOBAL CONVERGENCE

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A SCALED conjugate-gradient ALGORITHM FOR FAST SUPERVISED LEARNING

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NEURAL NETWORKS 1993年第4期6卷 525-533页

作者： MOLLER, MF University of Aarhus Denmark

A supervised learning algorithm (Scaled conjugate gradient, SCG) is introduced. The performance of SCG is benchmarked against that of the standard back propagation algorithm (BP) (Rumelhart, Hinton, & Williams, 1986), the conjugate gradient algorithm with line search (CGL) (Johansson, Dowla, & Goodman, 1990) and the one-step Broyden-Fletcher-Goldfarb-Shanno memoriless quasi-Newton algorithm (BFGS) (Battiti, 1990). SCG is fully-automated, includes no critical user-dependent parameters, and avoids a time consuming line search, which CGL and BFGS use in each iteration in order to determine an appropriate step size. Experiments show that SCG is considerably faster than BP, CGL, and BFGS.

关键词： FEEDFORWARD NEURAL NETWORK SUPERVISED LEARNING OPTIMIZATION conjugate gradient algorithms

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A Spectral RMIL plus conjugate gradient Method for Unconstrained Optimization With Applications in Portfolio Selection and Motion Control

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IEEE ACCESS 2021年 9卷 75398-75414页

作者： Awwal, Aliyu Muhammed Sulaiman, Ibrahim Mohammed Malik, Maulana Mamat, Mustafa Kumam, Poom Sitthithakerngkiet, Kanokwan King Mongkuts Univ Technol Thonburi KMUTT Ctr Excellence Theoret & Computat Sci TaCS CoE Fac Sci Bangkok 10140 Thailand Gombe State Univ Dept Math Fac Sci Gombe 760214 Nigeria Univ Sultan Zainal Abidin UniSZA Fac Informat & Comp Kuala Terengganu 22200 Malaysia Univ Indonesia UI Fac Math & Nat Sci Dept Math Depok 16424 Indonesia Univ Malaysia Perlis Inst Engn Math Arau 02600 Malaysia China Med Univ China Med Univ Hosp Dept Med Res Taichung 40402 Taiwan King Mongkuts Univ Technol North Bangkok KMUTNB Fac Appl Sci Dept Math Intelligent & Nonlinear Dynam Innovat Res Ctr Bangkok 10800 Thailand

The Spectral conjugate gradient (SCG) methods are among the efficient variants of CG algorithms which are obtained by combining the spectral gradient parameter and CG parameter. The success of SCG methods relies on effective choices of the step-size alpha(k) and the search direction d(k). This paper presents an SCG method for unconstrained optimization models. The search directions generated by the new method possess sufficient descent property without the restart condition and independent of the line search procedure used. The global convergence of the new method is proved under the weak Wolfe line search. Preliminary numerical results are presented which show that the method is efficient and promising, particularly for large-scale problems. Also, the method was applied to solve the robotic motion control problem and portfolio selection problem.

关键词： Spectral algorithm conjugate gradient algorithms unconstrained optimization models motion control line search procedure portfolio selection

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A Dai-Liao conjugate gradient algorithm with clustering of eigenvalues

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NUMERICAL algorithms 2018年第4期77卷 1273-1282页

作者： Andrei, Neculai Res Inst Informat Ctr Adv Modeling & Optimizat 8-10 Averescu Ave Bucharest 1 Romania Acad Romanian Scientists 54 Splaiul IndependenteiSect 5 Bucharest Romania

A new value for the parameter in Dai and Liao conjugate gradient algorithm is presented. This is based on the clustering of eigenvalues of the matrix which determine the search direction of this algorithm. This value of the parameter lead us to a variant of the Dai and Liao algorithm which is more efficient and more robust than the variants of the same algorithm based on minimizing the condition number of the matrix associated to the search direction. Global convergence of this variant of the algorithm is briefly discussed.

关键词： Unconstrained optimization conjugate gradient algorithms Eigenvalues clustering Condition number Wolfe conditions Convergence

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The constrained conjugate gradient algorithm

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IEEE SIGNAL PROCESSING LETTERS 2000年第12期7卷 351-354页

作者： Apolinario, JA de Campos, MLR Bernal, CP Escuela Politecn Ejercito Fac Ingn Elect Sangolqui Ecuador Univ Fed Rio de Janeiro COPPE Program Engn Eletr BR-21945970 Rio De Janeiro Brazil

Based on the condition for equivalence between linearly constrained minimum-variance (LCMV) filters and their generalized sidelobe canceler (GSC) implementations, we derive the new constrained conjugate gradient (CCG) algorithm. We discuss the use of orthogonal and nonorthogonal blocking matrices for the GSC structure and how the choice of this matrix may affect the relationship with the LCMV counterpart. The newly derived algorithm was tested in a computer experiment for adaptive multiuser detection and showed excellent results.

关键词： conjugate gradient algorithms constrained adaptive filtering

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Classification of normal and abnormal electrogastrograms using multilayer feedforward neural networks

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MEDICAL & BIOLOGICAL ENGINEERING & COMPUTING 1997年第3期35卷 199-206页

作者： Lin, Z Maris, J Hermans, L Vandewalle, J Chen, JDZ BAPTIST MED CTR LYNN INST HEALTHCARE RESOKLAHOMA CITYOK 73112 UNIV VIRGINIA HLTH SCI CTRCHARLOTTESVILLEVA KATHOLIEKE UNIV LEUVEN DEPT ELECT ENGNESATB-3001 LOUVAINBELGIUM

A neural network approach is proposed for the automated classification of the normal and abnormal EGG. Two learning algorithms, the quasi-Newton and the scaled conjugate gradient method for the multilayer feedforward neural networks (MFNN), are introduced and compared with the error backpropagation algorithm. The configurations of the MFNN are determined by experiment. The raw EGG data, its power spectral data, and its autoregressive moving average (ARMA) modelling parameters are used as the input to the MFNN and compared with each other. Three indexes (the percent correct, sum-squared error and complexity per iteration) are used to evaluate the performance of each learning algorithm. The results show that the scaled conjugate gradient algorithm performs best, in that it is robust and provides a super-linear convergence rate. The power spectral representation and the ARMA modelling parameters of the EGG are found to be better types of the input to the network for this specific application, both yielding a percent correctness of 95% on the test set. Although the results are focused on the classification of the EGG, this paper should provide useful information for the classification of other biomedical signals.

关键词： neural networks conjugate gradient algorithms pattern classification spectral analysis stomach electrogastrography

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