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On solving double direction methods for convex constrained monotone nonlinear equations with image restoration

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Abstract

Vast applications of derivative-free methods to restore the blurred images in compressive sensing has become an important trend in recent years. This research, a double direction method for better image restoration is proposed. Besides this, two double direction algorithms to solve constrained monotone nonlinear equations are presented. The main idea employed in the first algorithm is to approximate the Jacobian matrix via acceleration parameter to propose an effective derivative-free method. The second algorithm involve hybridizing the scheme of the first algorithm with Picard–Mann hybrid iterative scheme. In addition, the step length is calculated using inexact line search technique. The proposed methods are proven to be globally convergent under some mild conditions . The numerical experiment, shown in this paper, depicts the efficiency of the proposed methods. Furthermore, the second method is successfully applied to handle the \(\ell _1\)-norm regularization problem in image recovery which exhibits a better result than the existing method in the previous literature.

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References

  • Abdullahi H, Halilu AS, Waziri MY (2018) A modified conjugate gradient method via a double direction approach for solving large-scale symmetric nonlinear systems. J Numer Math Stoch 10(1):32–44

    MathSciNet  MATH  Google Scholar 

  • Awwal AM, Wang L, Kumam P, Mohammad H (2020) A two-step spectral gradient Projection method for system of nonlinear monotone equations and image deblurring problems. Symmetry 12:874. https://doi.org/10.3390/sym12060874

    Article  Google Scholar 

  • Dennis JE, More JJ (1974) A characterization of superlinear convergence and its application to quasi-newton methods. Math Comput 28:549–560

    Article  MathSciNet  Google Scholar 

  • Dennis JE, Schnabel RB (1983) Numerical methods for unconstrained optimization and non-linear equations. Prentice Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Dolan E, Moré J (2002) Benchmarking optimization software with performance profiles. J Math Program 91:201–213

    Article  MathSciNet  Google Scholar 

  • Duranovic-Milicic NI (2008) A multi step curve search algorithm in nonlinear optimization. Yugoslav J Oper Res 18:47–52

    Article  MathSciNet  Google Scholar 

  • Duranovic-Milicic NI, Gardasevic-Filipovic M (2010) A multi step curve search algorithm in nonlinear convex case. Facta Univ Ser Math Inform 25:11–24

    MathSciNet  MATH  Google Scholar 

  • Elaine T, Wotao Y, Yin Z (2007) A fixed-point continuation method for \(ell _1\)-regularized minimization with applications to compressed sensing. CAAM TR07-07, Rice University, pp 43–44

  • Figueiredo M, Nowak R, Wright SJ (2007) Gradient projection for sparse reconstruction, application to compressed sensing and other inverse problems. IEEE J-STSP IEEE Press, Piscataway, pp 586–597

    Google Scholar 

  • Halilu AS, Waziri MY, Yusuf I (2020) Efficient matrix-free direction method with line search for solving large-scale system of nonlinear equations. Yugoslav J Oper Res 30(4):399–412

  • Halilu AS, Waziri MY (2017) Enhanced matrix-free method via double step length approach for solving systems of nonlinear equations. Int J Appl Math Res 6:147–156

    Article  Google Scholar 

  • Halilu AS, Waziri MY (2017) A transformed double step length method for solving large-scale systems of nonlinear equations. J Numer Math Stoch 9:20–32

    MathSciNet  MATH  Google Scholar 

  • Halilu AS, Waziri MY (2018) An improved derivative-free method via double direction approach for solving systems of nonlinear equations. J Ramanujan Math Soc 33:75–89

    MathSciNet  MATH  Google Scholar 

  • Halilu AS, Waziri MY (2020) Inexact double step length method for solving systems of nonlinear equations. Stat Optim Inf Comput 8:165–174

    Article  MathSciNet  Google Scholar 

  • Halilu AS, Waziri MY (2020) Solving systems of nonlinear equations using improved double direction method. J Niger Math Soc 32(2):287–301

    MathSciNet  Google Scholar 

  • Halilu AS, Dauda MK, Waziri MY, Mamat M (2019) A derivative-free decent method via acceleration parameter for solving systems of nonlinear equations. Open J Sci Tech 2(3):1–4

    Article  Google Scholar 

  • Halilu AS, Majumder A, Waziri MY, Abdullahi H (2020) Double direction and step length method for solving system of nonlinear equations. Eur J Mol Clin Med 7(7):3899–3913

    Google Scholar 

  • Halilu AS, Majumder A, Waziri MY, Ahmed K (2021) Signal recovery with convex constrained nonlinear monotone equations through conjugate gradient hybrid approach. Math Comput Simul 187:520–539

    Article  MathSciNet  Google Scholar 

  • Ishikawa S (1974) Fixed points by a new iteration method. Proc Am Math Soc 44:147–150

    Article  MathSciNet  Google Scholar 

  • Kačurovskii RI (1960) On monotone operators and convex functionals. Usp Math Nauk 15(4):213–215

    Google Scholar 

  • Li D, Fukushima M (1999) A global and superlinear convergent Gauss-Newton based BFGS method for symmetric nonlinear equation. SIAM J Numer Anal 37:152–172

    Article  MathSciNet  Google Scholar 

  • Mann WR (1953) Mean value methods in iterations. Proc Am Math Soc 4:506–510

    Article  MathSciNet  Google Scholar 

  • Mario AT, Figueiredo R, Nowak D (2003) An EM algorithm for wavelet-based image restoration. IEEE Trans Image Process 12(8):906–916

    Article  MathSciNet  Google Scholar 

  • Meintjes K, Morgan AP (1987) A methodology for solving chemical equilibrium systems. Appl Math Comput 22:333–361

    MathSciNet  MATH  Google Scholar 

  • Minty GJ (1962) Monotone (nonlinear) operators in Hilbert space. Duke Math J 29(3):341–346

    Article  MathSciNet  Google Scholar 

  • Mohammad H (2020) A descent derivative-free algorithm for nonlinear monotone equations with convex constraints. RAIRO-Oper Res 54:489–505

    Article  MathSciNet  Google Scholar 

  • Mohammad H, Abubakar AB (2017) Barzilai–Borwein-like method for solving large-scale nonlinear systems of equations. J Niger Math Soc 36:71–83

    Google Scholar 

  • Pang JS (1986) Inexact Newton methods for the nonlinear complementarity problem. Math Program 1:54–71

    Article  MathSciNet  Google Scholar 

  • Petrović MJ, Stanimirović PS, Kontrec N, Mladenović J (2018) Hybrid modification of accelerated double direction method. Math Probl Eng Article ID 1523267, 8. https://doi.org/10.1155/2018/1523267

  • Petrović MJ, Stanimirovic PS (2014) Accelerated double direction method for solving unconstrained optimization problems. Math Probl Eng 2014:1–8

    Article  MathSciNet  Google Scholar 

  • Picard E (1890) Memoire sur la theorie des equations aux derivees partielles et la meth-ode des approximations successives. J Math Pures Appl 6:145–210

    MATH  Google Scholar 

  • Safeer HK (2013) A Picard–Mann hybrid iterative process. Fixed Point Theory Appl 69:10

    MathSciNet  MATH  Google Scholar 

  • Solodov MV, Svaiter BF (1999) A globally convergent inexact Newton method for systems of monotone equations. Reformulation: nonsmooth, piecewise smooth, semismooth and smoothing methods. Springer, Berlin, pp 355–369

    Google Scholar 

  • Sun M, Tian MY, Wang YJ (2019) Multi-step discrete-time Zhang neural networks with application to time-varying nonlinear optimization. Discret Dyn Nat Soc Article ID 4745759:1–14

    Google Scholar 

  • Tibshirani R (1996) Regression shrinkage and selection via the lasso. J R Stat Soc 58(1):267–288

    MathSciNet  MATH  Google Scholar 

  • Wang CW, Wang YJ (2009) A superlinearly convergent projection method for constrained systems of nonlinear equations. J Glob Optim 44(2):283–296

    Article  MathSciNet  Google Scholar 

  • Wang Z, Bovik AC, Sheikh HR, Simoncelli EP (2004) Image quality assessment: from error visibility to structural similarity. IEEE Trans Image Process 13:600–612

    Article  Google Scholar 

  • Wang C, Wang Y, Xu C (2007) A projection method for a system of nonlinear monotone equations with convex constraints. Math Methods Oper Res 66(1):33–46

    Article  MathSciNet  Google Scholar 

  • Waziri MY, Leong WJ, Hassan MA (2011) Jacobian-free diagonal Newton’s method for solving nonlinear systems with singular jacobian. Malay J Math Sci 5:241–255

  • Xiao YH, Zhu H (2013) A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J Math Anal Appl 405:310–319

    Article  MathSciNet  Google Scholar 

  • Xiao Y, Wang Q, Hu Q (2011) Non-smooth equations based method for \(\ell _1\)-norm problems with applications to compressed sensing. Nonlinear Anal Theory Methods Appl 74(11):3570–3577

    Article  Google Scholar 

  • Xiao Y, Wang Q, Hu Q (2011) Non-smooth equations based methods for \(l1-\)norm problems with applications to compressed sensing. Nonlinear Anal Theory Methods Appl 74:3570–3577

    Article  Google Scholar 

  • Yuan G, Lu X (2008) A new backtracking inexact BFGS method for symmetric nonlinear equations. Comput Math Appl 55:116–129

    Article  MathSciNet  Google Scholar 

  • Zarantonello EH (1960) Solving functional equations by contractive averaging. Tech. Rep. 160. U. S. Army Math. Research Center, Madison, Wisconsin

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Authors and Affiliations

  1. Department of Mathematics, School of Chemical Engineering and Physical Sciences, Lovely Professional University, Phagwara, 144411, India

    Abubakar Sani Halilu & Arunava Majumder

  2. Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, Nigeria

    Abubakar Sani Halilu

  3. Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano, Nigeria

    Mohammed Yusuf Waziri & Kabiru Ahmed

  4. Department of Mathematics, Faculty of Science, Gombe State University, Gombe, Nigeria

    Aliyu Muhammed Awwal

  5. KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok, 10140, Thailand

    Aliyu Muhammed Awwal

  6. Numerical Optimization Research Group, Bayero University, Kano, Nigeria

    Abubakar Sani Halilu, Mohammed Yusuf Waziri, Aliyu Muhammed Awwal & Kabiru Ahmed

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  1. Abubakar Sani Halilu
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  2. Arunava Majumder
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  3. Mohammed Yusuf Waziri
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  4. Aliyu Muhammed Awwal
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  5. Kabiru Ahmed
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Correspondence to Abubakar Sani Halilu.

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Halilu, A.S., Majumder, A., Waziri, M.Y. et al. On solving double direction methods for convex constrained monotone nonlinear equations with image restoration. Comp. Appl. Math. 40, 239 (2021). https://doi.org/10.1007/s40314-021-01624-1

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  • DOI: https://doi.org/10.1007/s40314-021-01624-1

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