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A note on the spectral gradient projection method for nonlinear monotone equations with applications

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Abstract

In this work, we provide a note on the spectral gradient projection method for solving nonlinear equations. Motivated by recent extensions of the spectral gradient method for solving nonlinear monotone equations with convex constraints, in this paper, we note that choosing the search direction as a convex combination of two different positive spectral coefficients multiplied with the residual vector is more efficient and robust compared with the standard choice of spectral gradient coefficients combined with the projection strategy of Solodov and Svaiter (A globally convergent inexact newton method for systems of monotone equations. In: Reformulation: Nonsmooth. Piecewise Smooth, Semismooth and Smoothing Methods, pp 355–369. Springer, 1998). Under suitable assumptions, the convergence of the proposed method is established. Preliminary numerical experiments show that the method is promising. In this paper, the proposed method was used to recover sparse signal and restore blurred image arising from compressive sensing.

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References

  • Abubakar AB, Kumam P, Awwal AM, Thounthong P (2019a) A modified self-adaptive conjugate gradient method for solving convex constrained monotone nonlinear equations for signal reovery problems. Mathematics 7(8):693. https://doi.org/10.3390/math7080693

    Article  Google Scholar 

  • Abubakar AB, Kumam P, Mohammad H, Awwal AM (2019b) An efficient conjugate gradient method for convex constrained monotone nonlinear equations with applications. Mathematics 7(9):767. https://doi.org/10.3390/math7090767

    Article  Google Scholar 

  • Abubakar AB, Kumam P, Mohammad H, Awwal AM, Sitthithakerngkiet K (2019c) A modified fletcher-reeves conjugate gradient method for monotone nonlinear equations with some applications. Mathematics 7(8):745

    Article  Google Scholar 

  • Barzilai J, Borwein JM (1988) Two-point step size gradient methods. IMA J. Numer. Anal. 8(1):141–148

    Article  MathSciNet  MATH  Google Scholar 

  • Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1):183–202

    Article  MathSciNet  MATH  Google Scholar 

  • Bellavia S, Macconi M, Morini B (2004) Strscne: a scaled trust-region solver for constrained nonlinear equations. Comput. Optim. Appl. 28(1):31–50

    Article  MathSciNet  MATH  Google Scholar 

  • Bing Y, Lin G (1991) An efficient implementation of Merrill’s method for sparse or partially separable systems of nonlinear equations. SIAM J. Optim. 1(2):206–221. https://doi.org/10.1137/0801015

    Article  MathSciNet  MATH  Google Scholar 

  • Bruckstein AM, Donoho DL, Elad M (2009) From sparse solutions of systems of equations to sparse modeling of signals and images. SIAM Rev. 51(1):34–81

    Article  MathSciNet  MATH  Google Scholar 

  • Dirkse FMSP (1995) A collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5:319–345

    Article  Google Scholar 

  • Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math. Program. 91(2):201–213

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Figueiredo MA, Nowak RD, Wright SJ (2007) Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4):586–597

    Article  Google Scholar 

  • Fukushima M (1992) Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Math. Program. 53(1–3):99–110

    Article  MathSciNet  MATH  Google Scholar 

  • Ghaddar B, Marecek J, Mevissen M (2016) Optimal power flow as a polynomial optimization problem. IEEE Trans. Power Syst. 31(1):539–546

    Article  Google Scholar 

  • Hager W, Zhang H (2005) A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16(1):170–192. https://doi.org/10.1137/030601880

    Article  MathSciNet  MATH  Google Scholar 

  • Hager WW, Phan DT, Zhang H (2011) Gradient-based methods for sparse recovery. SIAM J. Imaging Sci. 4(1):146–165

    Article  MathSciNet  MATH  Google Scholar 

  • Hale ET, Yin W, Zhang Y (2007) A fixed-point continuation method for \(\ell _1\)-regularized minimization with applications to compressed sensing. CAAM TR07-07, Rice University 43:44

  • Iusem NA, Solodov VM (1997) Newton-type methods with generalized distances for constrained optimization. Optimization 41(3):257–278

    Article  MathSciNet  MATH  Google Scholar 

  • Kanzow C, Yamashita N, Fukushima M (2004) Levenberg-marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints. J. Comput. Appl. Math. 172(2):375–397

    Article  MathSciNet  MATH  Google Scholar 

  • La Cruz W, Martínez J, Raydan M (2006) Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75(255):1429–1448

    Article  MathSciNet  MATH  Google Scholar 

  • Liu J, Li SJ (2015) A projection method for convex constrained monotone nonlinear equations with applications. Comput. Math. Appl. 70(10):2442–2453

    Article  MathSciNet  Google Scholar 

  • Meintjes K, Morgan AP (1987) A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22(4):333–361

    MathSciNet  MATH  Google Scholar 

  • Mohammad H, Abubakar AB (2017) A positive spectral gradient-like method for large-scale nonlinear monotone equations. Bull. Comput. Appl. Math. 5(1):99–115

    MathSciNet  MATH  Google Scholar 

  • Raydan M (1997) The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem. SIAM J. Optim. 7(1):26–33

    Article  MathSciNet  MATH  Google Scholar 

  • Solodov MV, Svaiter BF (1998) A globally convergent inexact newton method for systems of monotone equations. In: Reformulation: Nonsmooth. Piecewise Smooth, Semismooth and Smoothing Methods, pp 355–369. Springer

  • Van Den Berg E, Friedlander MP (2008) Probing the pareto frontier for basis pursuit solutions. SIAM J. Sci. Comput. 31(2):890–912

    Article  MathSciNet  MATH  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(4):600–612

    Article  Google Scholar 

  • Wood AJ, Wollenberg BF (2012) Power Generation, Operation, and Control. Wiley, New York

    Google Scholar 

  • Xiao Y, Zhu H (2013) A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405(1):310–319

    Article  MathSciNet  MATH  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  MATH  Google Scholar 

  • Yamashita N, Fukushima M (1997) Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems. Math. Program. 76(3):469–491. https://doi.org/10.1007/BF02614394

    Article  MathSciNet  MATH  Google Scholar 

  • Yu Z, Lin J, Sun J, Xiao YH, Liu L, Li ZH (2009) Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59(10):2416–2423

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang L, Zhou W (2006) Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196(2):478–484

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao Y, Li D (2001) Monotonicity of fixed point and normal mappings associated with variational inequality and its application. SIAM J. Optim. 11(4):962–973

    Article  MathSciNet  MATH  Google Scholar 

  • Zhou WJ, Li DH (2008) A globally convergent BFGS method for nonlinear monotone equations without any merit functions. Math. Comput. 77(264):2231–2240

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referees for helpful suggestions that improved the paper. We are indebted to Associate Professor Mark Dougherty affiliated at the Auburn University, USA, for his proofreading and helpful comments. We would like to thank Professor Jinkui Liu affiliated at the Chongqing Three Gorges University, Chongqing, China, for providing us access to the MATLAB source codes for SGCS and PCG methods. The first author was supported by the Petchra Pra Jom Klao Doctoral Scholarship Academic for Ph.D. Program at KMUTT (2017–2020).

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Correspondence to Auwal Bala Abubakar.

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Communicated by Natasa Krejic.

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Abubakar, A.B., Kumam, P. & Mohammad, H. A note on the spectral gradient projection method for nonlinear monotone equations with applications. Comp. Appl. Math. 39, 129 (2020). https://doi.org/10.1007/s40314-020-01151-5

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