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A framework for convex-constrained monotone nonlinear equations and its special cases

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Abstract

This work refers to methods for solving convex-constrained monotone nonlinear equations. We first propose a framework, which is obtained by combining a safeguard strategy on the search directions with a notion of approximate projections. The global convergence of the framework is established under appropriate assumptions and some examples of methods which fall into this framework are presented. In particular, inexact versions of steepest descent-based, spectral gradient-like, Newton-like and limited memory BFGS methods are discussed. Numerical experiments illustrating the practical behavior of the algorithms are discussed and comparisons with existing methods are also presented.

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References

  • Abubakar AB, Kumam P, Mohammad H (2020) A note on the spectral gradient projection method for nonlinear monotone equations with applications. Comp. Appl. Math. 39(2):129

    Article  MathSciNet  MATH  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 

  • Cottle RW, Pang J-S, Stone RE (2009) The Linear Complementarity Problem. Society for Industrial and Applied Mathematics

  • Cottle RW, Dantzig GB (1968) Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1(1):103–125

    Article  MathSciNet  MATH  Google Scholar 

  • Cruz JYB, Ferreira OP, Prudente LF (2016) On the global convergence of the inexact semi-smooth Newton method for absolute value equation. Comput. Opt. Appl. 65(1):93–108

    Article  MathSciNet  MATH  Google Scholar 

  • Dai Y-H, Al-Baali M, Yang X (2015) A positive Barzilai-Borwein-like stepsize and an extension for symmetric linear systems. In: Numer. Anal. Optim. vol. 134, pp. 59–75. Springer

  • Dirkse SP, Ferris MC (1995) MCPLIB: a collection of nonlinear mixed complementarity problems. Opt. Meth. Softw. 5(4):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 

  • Dunn JC (1980) Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM J. Control. Opt. 18(5):473–487

    Article  MathSciNet  MATH  Google Scholar 

  • El-Hawary ME (1996) Optimal Power Flow: Solution Techniques, Requirements and Challenges: IEEE Tutorial Course. Piscataway: IEEE

  • Frank M, Wolfe P (1956) An algorithm for quadratic programming. Nav. Res. Logist. Q. 3(1–2):95–110

    Article  MathSciNet  Google Scholar 

  • La Cruz W (2017) A spectral algorithm for large-scale systems of nonlinear monotone equations. Numer. Algor. 76(4):1109–1130

    Article  MathSciNet  MATH  Google Scholar 

  • La Cruz W, Raydan M (2003) Nonmonotone spectral methods for large-scale nonlinear systems. Opt. Meth. Softw. 18(5):583–599

    Article  MathSciNet  MATH  Google Scholar 

  • Liu J, Feng Y (2019) A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algor. 82(1):245–262

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Mangasarian OL (2009) A generalized Newton method for absolute value equations. Opt. Lett. 3(1):101–108

    Article  MathSciNet  MATH  Google Scholar 

  • Mangasarian OL, Meyer RR (2006) Absolute value equations. Linear Algebra Appl. 419(2–3):359–367

    Article  MathSciNet  MATH  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 (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 

  • Oliveira FR, Ferreira OP (2020) Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations. Appl. Numer. Math. 156:63–76

    Article  MathSciNet  MATH  Google Scholar 

  • Ou Y, Liu Y (2017) Supermemory gradient methods for monotone nonlinear equations with convex constraints. Comp. Appl. Math. 36(1):259–279

    Article  MathSciNet  MATH  Google Scholar 

  • Solodov MV, Svaiter BF (1999) In: Fukushima, M., Qi, L. (eds.) A Globally Convergent Inexact Newton Method for Systems of Monotone Equations, pp. 355–369. Springer, Boston, MA

  • Solodov MV, Svaiter BF (1999) A new projection method for variational inequality problems. SIAM J. Control Opt. 37(3):765–776

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Wood AJ, Wollenberg BF (1996) Power generation operation and control, published by John Wiley and Sons. New York, January

  • Yu Z, Lin J, Sun J, Xiao Y, Liu L, Li Z (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 

  • Zhou W, Li D (2007) Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 25(1):89–96

    MathSciNet  Google Scholar 

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Funding

The work of these authors was supported in part by CAPES and CNPq Grants 405349/2021-1 and 304133/2021-3.

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Authors and Affiliations

  1. IME, Universidade Federal de Goiás, R. Jacarandá, Goiânia, 74001-970, Goiás, Brazil

    Max L. N. Gonçalves & Tiago C. Menezes

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  1. Max L. N. Gonçalves
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  2. Tiago C. Menezes
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Correspondence to Max L. N. Gonçalves.

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Communicated by Carlos Conca.

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Gonçalves, M.L.N., Menezes, T.C. A framework for convex-constrained monotone nonlinear equations and its special cases. Comp. Appl. Math. 42, 306 (2023). https://doi.org/10.1007/s40314-023-02446-z

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  • DOI: https://doi.org/10.1007/s40314-023-02446-z

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