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
Barzilai J, Borwein JM (1988) Two-point step size gradient methods. IMA J. Numer. Anal. 8(1):141–148
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
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
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
Dolan ED, Moré JJ (2002) Benchmarking optimization software with performance profiles. Math. Program. 91(2):201–213
Dunn JC (1980) Convergence rates for conditional gradient sequences generated by implicit step length rules. SIAM J. Control. Opt. 18(5):473–487
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
La Cruz W (2017) A spectral algorithm for large-scale systems of nonlinear monotone equations. Numer. Algor. 76(4):1109–1130
La Cruz W, Raydan M (2003) Nonmonotone spectral methods for large-scale nonlinear systems. Opt. Meth. Softw. 18(5):583–599
Liu J, Feng Y (2019) A derivative-free iterative method for nonlinear monotone equations with convex constraints. Numer. Algor. 82(1):245–262
Liu JK, Li SJ (2015) A projection method for convex constrained monotone nonlinear equations with applications. Comput. Math. Appl. 70(10):2442–2453
Mangasarian OL (2009) A generalized Newton method for absolute value equations. Opt. Lett. 3(1):101–108
Mangasarian OL, Meyer RR (2006) Absolute value equations. Linear Algebra Appl. 419(2–3):359–367
Meintjes K, Morgan AP (1987) A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22(4):333–361
Mohammad H (2017) A positive spectral gradient-like method for large-scale nonlinear monotone equations. Bull. Comput. Appl. Math. 5(1):99–115
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
Ou Y, Liu Y (2017) Supermemory gradient methods for monotone nonlinear equations with convex constraints. Comp. Appl. Math. 36(1):259–279
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
Wang C, Wang Y (2009) A superlinearly convergent projection method for constrained systems of nonlinear equations. J. Glob. Opt. 44(2):283–296
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
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
Zhang L, Zhou W (2006) Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196(2):478–484
Zhou W, Li D (2007) Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 25(1):89–96
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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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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
Keywords
- Monotone nonlinear equations
- Approximate projection
- Global convergence
- Steepest descent-based method
- Spectral gradient-like method
- Newton-like method