An efficient DY-type spectral conjugate gradient method for system of nonlinear monotone equations with application in signal recovery

Many problems in engineering and social sciences can be transformed into system of nonlinear equations. As a result, a lot of methods have been proposed for solving the system. Some of the classical methods include Newton and Quasi Newton methods which have rapid convergence from good initial points but unable to deal with large scale problems due to the computation of Jacobian matrix or its approximation. Spectral and conjugate gradient methods proposed for unconstrained optimization, and later on extended to solve nonlinear equations do not require any computation of Jacobian matrix or its approximation, thus, are suitable to handle large scale problems. In this paper, we proposed a spectral conjugate gradient algorithm for solving system of nonlinear equations where the operator under consideration is monotone. The search direction of the proposed algorithm is constructed by taking the convex combination of the Dai-Yuan (DY) parameter and a modified conjugate descent (CD) parameter. The proposed search direction is sufficiently descent and under some suitable assumptions, the global convergence of the proposed algorithm is proved. Numerical experiments on some test problems are presented to show the efficiency of the proposed algorithm in comparison with an existing one. Finally, the algorithm is successfully applied in signal recovery problem arising from compressive sensing.

[1]  Robert D. Nowak,et al.  An EM algorithm for wavelet-based image restoration , 2003, IEEE Trans. Image Process..

[2]  SunJing,et al.  Spectral gradient projection method for monotone nonlinear equations with convex constraints , 2009 .

[3]  Chuanwei Wang,et al.  A projection method for a system of nonlinear monotone equations with convex constraints , 2007, Math. Methods Oper. Res..

[4]  Michael P. Friedlander,et al.  Probing the Pareto Frontier for Basis Pursuit Solutions , 2008, SIAM J. Sci. Comput..

[5]  Aliyu Muhammed Awwal,et al.  A Perry-type derivative-free algorithm for solving nonlinear system of equations and minimizing ℓ1 regularized problem , 2020 .

[6]  K. Toh,et al.  Superlinear Convergence of a Newton-Type Algorithm for Monotone Equations , 2005 .

[7]  S. Dirkse,et al.  Mcplib: a collection of nonlinear mixed complementarity problems , 1995 .

[8]  Jorge J. Moré,et al.  Digital Object Identifier (DOI) 10.1007/s101070100263 , 2001 .

[9]  Poom Kumam,et al.  Inertial-Based Derivative-Free Method for System of Monotone Nonlinear Equations and Application , 2020, IEEE Access.

[10]  Jong-Shi Pang,et al.  Inexact Newton methods for the nonlinear complementarity problem , 1986, Math. Program..

[11]  Duan Li,et al.  Monotonicity of Fixed Point and Normal Mappings Associated with Variational Inequality and Its Application , 2000, SIAM J. Optim..

[12]  Dong-Hui Li,et al.  A globally convergent BFGS method for nonlinear monotone equations without any merit functions , 2008, Math. Comput..

[13]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[14]  Poom Kumam,et al.  Spectral modified Polak-Ribiére-Polyak projection conjugate gradient method for solving monotone systems of nonlinear equations , 2019, Appl. Math. Comput..

[15]  M. V. Solodovy,et al.  Newton-type Methods with Generalized Distances For Constrained Optimization , 1997 .

[16]  Miss A.O. Penney (b) , 1974, The New Yale Book of Quotations.

[17]  Yunhai Xiao,et al.  Spectral gradient projection method for monotone nonlinear equations with convex constraints , 2009 .

[18]  M. Fukushima Title Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality , 2018 .

[19]  Zhong Wan,et al.  A Modified Spectral PRP Conjugate Gradient Projection Method for Solving Large-Scale Monotone Equations and Its Application in Compressed Sensing , 2019, Mathematical Problems in Engineering.

[20]  Shengjie Li,et al.  A projection method for convex constrained monotone nonlinear equations with applications , 2015, Comput. Math. Appl..

[21]  Masao Fukushima,et al.  A Globally and Superlinearly Convergent Gauss-Newton-Based BFGS Method for Symmetric Nonlinear Equations , 1999, SIAM J. Numer. Anal..

[22]  Mikhail V. Solodov,et al.  A Globally Convergent Inexact Newton Method for Systems of Monotone Equations , 1998 .

[23]  Masao Fukushima,et al.  Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems , 1992, Math. Program..

[24]  Yunhai Xiao,et al.  A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing , 2013 .

[25]  Allen J. Wood,et al.  Power Generation, Operation, and Control , 1984 .

[26]  Wotao Yin,et al.  TR 0707 A Fixed-Point Continuation Method for ` 1-Regularized Minimization with Applications to Compressed Sensing , 2007 .

[27]  Poom Kumam,et al.  A modified conjugate gradient method for monotone nonlinear equations with convex constraints , 2019, Applied Numerical Mathematics.

[28]  Li,et al.  Spectral DY-Type Projection Method for Nonlinear Monotone Systems of Equations , 2015 .

[29]  Jing Liu,et al.  Two Improved Conjugate Gradient Methods with Application in Compressive Sensing and Motion Control , 2020 .

[30]  J. J. Moré,et al.  A Characterization of Superlinear Convergence and its Application to Quasi-Newton Methods , 1973 .

[31]  Wanyou Cheng,et al.  A PRP type method for systems of monotone equations , 2009, Math. Comput. Model..

[32]  Yunhai Xiao,et al.  Non-smooth equations based method for l 1 -norm problems with applications to compressed sensing , 2011 .

[33]  Shengjie Li,et al.  Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations , 2016 .

[34]  Poom Kumam,et al.  A structured quasi-Newton algorithm with nonmonotone search strategy for structured NLS problems and its application in robotic motion control , 2021, J. Comput. Appl. Math..

[35]  José Mario Martínez,et al.  Nonmonotone Spectral Projected Gradient Methods on Convex Sets , 1999, SIAM J. Optim..

[36]  Hassan Mohammad,et al.  A Barzilai-Borwein gradient projection method for sparse signal and blurred image restoration , 2020, J. Frankl. Inst..

[37]  Ya-Xiang Yuan,et al.  A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property , 1999, SIAM J. Optim..

[38]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[39]  Poom Kumam,et al.  A Modified Conjugate Descent Projection Method for Monotone Nonlinear Equations and Image Restoration , 2020, IEEE Access.

[40]  P. Alam ‘G’ , 2021, Composites Engineering: An A–Z Guide.

[41]  A. Morgan,et al.  A methodology for solving chemical equilibrium systems , 1987 .

[42]  José Mario Martínez,et al.  Spectral residual method without gradient information for solving large-scale nonlinear systems of equations , 2006, Math. Comput..

[43]  Hassan Mohammad,et al.  A Projection Hestenes–Stiefel Method with Spectral Parameter for Nonlinear Monotone Equations and Signal Processing , 2020, Mathematical and Computational Applications.

[44]  Yang Bing,et al.  AN EFFICIENT IMPLEMENTATION OF MERRILL'S METHOD FOR SPARSE OR PARTIALLY SEPARABLE SYSTEMS OF NONLINEAR EQUATIONS* , 1991 .

[45]  Poom Kumam,et al.  Two Hybrid Spectral Methods With Inertial Effect for Solving System of Nonlinear Monotone Equations With Application in Robotics , 2021, IEEE Access.

[46]  Hassan Mohammad,et al.  A Two-Step Spectral Gradient Projection Method for System of Nonlinear Monotone Equations and Image Deblurring Problems , 2020, Symmetry.

[47]  William La Cruz A spectral algorithm for large-scale systems of nonlinear monotone equations , 2017, Numerical Algorithms.

[48]  Poom Kumam,et al.  An efficient gradient-free projection algorithm for constrained nonlinear equations and image restoration , 2021 .

[49]  Yuming Feng,et al.  A derivative-free iterative method for nonlinear monotone equations with convex constraints , 2018, Numerical Algorithms.

[50]  Hassan Mohammad,et al.  A note on the spectral gradient projection method for nonlinear monotone equations with applications , 2020, Comput. Appl. Math..