A modified Liu-Storey-Conjugate descent hybrid projection method for convex constrained nonlinear equations and image restoration

We present an iterative method for solving the convex constraint nonlinear equation problem. The method incorporates the projection strategy by Solodov and Svaiter with the hybrid Liu-Storey and Conjugate descent method by Yang et al. for solving the unconstrained optimization problem. The proposed method does not require the Jacobian information, nor does it require to store any matrix at each iteration. Thus, it has the potential to solve large-scale non-smooth problems. Under some standard assumptions, the convergence analysis of the method is established. Finally, to show the applicability of the proposed method, the proposed method is used to solve the \begin{document}$ \ell_1 $\end{document} -norm regularized problems to restore blurred and noisy images. The numerical experiment indicates that our result is a significant improvement compared with the related methods for solving the convex constraint nonlinear equation problem.

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

[2]  Weijun Zhou,et al.  Spectral gradient projection method for solving nonlinear monotone equations , 2006 .

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

[4]  Hassan Mohammad,et al.  A descent derivative-free algorithm for nonlinear monotone equations with convex constraints , 2020, RAIRO Oper. Res..

[5]  Poom Kumam,et al.  A Family of Derivative-Free Conjugate Gradient Methods for Constrained Nonlinear Equations and Image Restoration , 2020, IEEE Access.

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

[7]  Spectral Conjugate Gradient Like Method for Signal Reconstruction , 2020 .

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

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

[10]  Yunhai Xiao,et al.  A class of conjugate gradient methods for convex constrained monotone equations , 2017 .

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

[12]  Jorge J. Moré,et al.  Benchmarking optimization software with performance profiles , 2001, Math. Program..

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

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

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

[16]  Abdulkarim Hassan Ibrahim,et al.  Derivative-free HS-DY-type method for solving nonlinear equations and image restoration , 2020, Heliyon.

[17]  Seyed Mehdi Lajevardi,et al.  Structural similarity classifier for facial expression recognition , 2014, Signal Image Video Process..

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

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

[20]  Marcos Raydan,et al.  Nonmonotone Spectral Methods for Large-Scale Nonlinear Systems , 2003, Optim. Methods Softw..

[21]  P. Kumam,et al.  An improved three-term derivative-free method for solving nonlinear equations , 2018, Computational and Applied Mathematics.

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

[23]  Abdulkarim Hassan Ibrahim,et al.  FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations , 2021 .

[24]  Poom Kumam,et al.  A hybrid conjugate gradient algorithm for constrained monotone equations with application in compressive sensing , 2020, Heliyon.

[25]  Poom Kumam,et al.  Least-Square-Based Three-Term Conjugate Gradient Projection Method for ℓ1-Norm Problems with Application to Compressed Sensing , 2020, Mathematics.

[26]  Hassan Mohammad,et al.  PRP-like algorithm for monotone operator equations , 2021, Japan Journal of Industrial and Applied Mathematics.

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

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

[29]  Poom Kumam,et al.  A descent Dai-Liao conjugate gradient method for nonlinear equations , 2018, Numerical Algorithms.

[30]  M. Solodov,et al.  A New Projection Method for Variational Inequality Problems , 1999 .

[31]  Abdulkarim Hassan Ibrahim,et al.  A New Three-Term Hestenes-Stiefel Type Method for Nonlinear Monotone Operator Equations and Image Restoration , 2021, IEEE Access.

[32]  Kanikar Muangchoo,et al.  Derivative-free SMR conjugate gradient method for constraint nonlinear equations , 2021 .

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

[34]  Poom Kumam,et al.  Re-modified derivative-free iterative method for nonlinear monotone equations with convex constraints , 2021 .

[35]  Boris Polyak.  The conjugate gradient method in extremal problems , 1969 .

[36]  Hassan Mohammad,et al.  A Modified Fletcher–Reeves Conjugate Gradient Method for Monotone Nonlinear Equations with Some Applications , 2019, Mathematics.

[37]  Zhijun Luo,et al.  A Global Convergence of LS-CD Hybrid Conjugate Gradient Method , 2013, Adv. Numer. Anal..

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

[39]  Hassan Mohammad.  BARZILAI-BORWEIN-LIKE METHOD FOR SOLVING LARGE-SCALE NON-LINEAR SYSTEMS OF EQUATIONS , 2017 .

[40]  C. Chidume,et al.  Iterative algorithms for solutions of Hammerstein equations in real Banach spaces , 2020 .