Top

Journal of Applied Mathematics and Computing

Published in:

01-02-2016 | Original Research

Two globally convergent nonmonotone trust-region methods for unconstrained optimization

Authors: Masoud Ahookhosh, Susan Ghaderi

Published in: Journal of Applied Mathematics and Computing | Issue 1-2/2016

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

This paper addresses some trust-region methods equipped with nonmonotone strategies for solving nonlinear unconstrained optimization problems. More specifically, the importance of using nonmonotone techniques in nonlinear optimization is motivated, then two new nonmonotone terms are proposed, and their combinations into the traditional trust-region framework are studied. The global convergence to first- and second-order stationary points and local superlinear and quadratic convergence rates for both algorithms are established. Numerical experiments on the CUTEst test collection of unconstrained problems and some highly nonlinear test functions are reported, where a comparison among state-of-the-art nonmonotone trust-region methods shows the efficiency of the proposed nonmonotone schemes.

previous article Maximum values of atom–bond connectivity index in the class of tricyclic graphs

next article Hopf bifurcation analysis for an SIRS epidemic model with logistic growth and delays

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Ahookhosh, M., Amini, K.: An efficient nonmonotone trust-region method for unconstrained optimization. Numer. Algorithms 59(4), 523–540 (2012)MathSciNetCrossRefMATH

Ahookhosh, M., Amini, K.: A nonmonotone trust region method with adaptive radius for unconstrained optimization problems. Comput. Math. Appl. 60(3), 411–422 (2010)MathSciNetCrossRefMATH

Ahookhosh, M., Amini, K., Bahrami, S.: A class of nonmonotone Armijo-type line search method for unconstrained optimization. Optimization 61(4), 387–404 (2012)MathSciNetCrossRefMATH

Ahookhosh, M., Amini, K., Peyghami, M.R.: A nonmonotone trust-region line search method for large-scale unconstrained optimization. Appl. Math. Model. 36(1), 478–487 (2012)MathSciNetCrossRefMATH

Ahookhosh, M., Esmaeili, H., Kimiaei, M.: An effective trust-region-based approach for symmetric nonlinear systems. Int. J. Comput. Math. 90(3), 671–690 (2013)CrossRefMATH

Ahookhosh, M., Ghederi, S.: On efficiency of nonmonotone Armijo-type line searches. http://arxiv.org/pdf/1408.2675 (2015)

Amini, K., Ahookhosh, M.: A hybrid of adjustable trust-region and nonmonotone algorithms for unconstrained optimization. Appl. Math. Model. 38, 2601–2612 (2014)MathSciNetCrossRef

Amini, K., Ahookhosh, M., Nosratipour, H.: An inexact line search approach using modified nonmonotone strategy for unconstrained optimization. Numer. Algorithms 66, 49–78 (2014)MathSciNetCrossRefMATH

Andrei, N.: An unconstrained optimization test functions collection. Adv. Model. Optim. 10(1), 147–161 (2008)MathSciNetMATH

10.

Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)MathSciNetCrossRefMATH

11.

Birgin, E.G., Martínez, J.M., Raydan, M.: Inexact spectral projected gradient methods on convex sets. IMA J. Numer. Anal. 23, 539–559 (2003)MathSciNetCrossRefMATH

12.

Bonnans, J.F., Panier, E., Tits, A., Zhou, J.L.: Avoiding the Maratos effect by means of a nonmonotone line search, II: inequality constrained problems—feasible iterates. SIAM J. Numer. Anal. 29, 1187–1202 (1992)MathSciNetCrossRefMATH

13.

Chamberlain, R.M., Powell, M.J.D., Lemarechal, C., Pedersen, H.C.: The watchdog technique for forcing convergence in algorithm for constrained optimization. Math. Progr. Study 16, 1–17 (1982)MathSciNetCrossRefMATH

14.

Conn, A.R., Gould, N.I.M., Toint, PhL: Trust-Region Methods, Society for Industrial and Applied Mathematics. SIAM, Philadelphia (2000)CrossRef

15.

Davidon, W.C.: Conic approximation and collinear scaling for optimizers. SIAM J. Numer. Anal. 17, 268–281 (1980)MathSciNetCrossRefMATH

16.

Deng, N.Y., Xiao, Y., Zhou, F.J.: Nonmonotonic trust region algorithm. J. Optim. Theory Appl. 76, 259–285 (1993)MathSciNetCrossRefMATH

17.

Dennis, J.E., Li, S.B., Tapia, R.A.: A unified approach to global convergence of trust region methods for nonsmooth optimization. Math. Progr. 68, 319–346 (1995)MathSciNetMATH

18.

Di, S., Sun, W.: Trust region method for conic model to solve unconstrained optimization problems. Optim. Methods Softw. 6, 237–263 (1996)CrossRef

19.

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

20.

Erway, J.B., Gill, P.E.: A subspace minimization method for the trust-region step. SIAM J. Optim. 20(3), 1439–1461 (2009)MathSciNetCrossRefMATH

21.

Erway, J.B., Gill, P.E., Griffin, J.D.: Iterative methods for finding a trust-region step. SIAM J. Optim. 20(2), 1110–1131 (2009)MathSciNetCrossRefMATH

22.

Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, New York (1987)MATH

23.

Ferreira, P.S., Karas, E.W., Sachine, M.: A globally convergent trust-region algorithm for unconstrained derivative-free optimization. Comput. Appl. Math. (2014). doi:10.1007/s40314-014-0167-2

24.

Grapiglia, G.N., Yuan, J., Yuan, Y.: A derivative-free trust-region algorithm for composite nonsmooth optimization. Comput. Appl. Math. (2014). doi:10.1007/s40314-014-0201-4

25.

Gould, N.I.M., Orban, D., Toint, P.H.L.: CUTEst: a constrained and unconstrained testing environment with safe threads for mathematical optimization. Comput. Optim. Appl. (2014). doi:10.1007/s10589-014-9687-3

26.

Gould, N.I.M., Lucidi, S., Roma, M., Toint, PhL: Solving the trust-region subproblem using the Lanczos method. SIAM J. Optim. 9(2), 504–525 (1999)MathSciNetCrossRefMATH

27.

Gould, N.I.M., Orban, D., Sartenaer, A., Toint, PhL: Sensitivity of trust-region algorithms to their parameters. Q. J. Oper. Res. 3, 227–241 (2005)MathSciNetCrossRefMATH

28.

Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)MathSciNetCrossRefMATH

29.

Gürbüzbalaban, M., Overton, M.L.: On Nesterov’s nonsmooth Chebyshev–Rosenbrock functions. Nonlinear Anal. 75, 1282–1289 (2012)MathSciNetCrossRefMATH

30.

Hallabi, M.E., Tapia, R.: A global convergence theory for arbitraty norm trust region methods for nonlinear equations. Mathematical Sciences Technical Report, TR 87–25, Rice University, Houston, TX (1987)

31.

Mo, J., Liu, C., Yan, S.: A nonmonotone trust region method based on nonincreasing technique of weighted average of the succesive function value. J. Comput. Appl. Math. 209, 97–108 (2007)MathSciNetCrossRefMATH

32.

Moré, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3), 553–572 (1983)CrossRefMATH

33.

Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, NewYork (2006)MATH

34.

Panier, E.R., Tits, A.L.: Avoiding the Maratos effect by means of a nonmonotone linesearch. SIAM J. Numer. Anal. 28, 1183–1195 (1991)MathSciNetCrossRefMATH

35.

Rojas, M., Sorensen, D.C.: A trust-region approach to the regularization of large-scale discrete forms of ill-posed problems. SIAM J. Sci. Comput. 26, 1842–1860 (2002)MathSciNetCrossRef

36.

Schultz, G.A., Schnabel, R.B., Byrd, R.H.: A family of trust-region-based algorithms for unconstrained minimization with strong global convergence. SIAM J. Numer. Anal. 22, 47–67 (1985)MathSciNetCrossRef

37.

Sorensen, D.C.: The \(q\)-superlinear convergence of a collinear scaling algorithm for unconstrained optimization. SIAM J. Numer. Anal. 17, 84–114 (1980)MathSciNetCrossRefMATH

38.

Sorensen, D.C.: Newton’s method with a model trust region modification. SIAM J. Numer. Anal. 19(2), 409–426 (1982)MathSciNetCrossRefMATH

39.

Steihaug, T.: The conjugate gradient method and trust regions in large scale optimization. SIAM J. Numer. Anal. 20(3), 626–637 (1983)MathSciNetCrossRefMATH

40.

Sun, W.: On nonquadratic model optimization methods. Asia Pac. J. Oper. Res. 13, 43–63 (1996)MATH

41.

Toint, Ph.L: An assessment of nonmonotone linesearch technique for unconstrained optimization. SIAM J. Sci. Comput. 17, 725–739 (1996)

42.

Toint, Ph.L: Non-monotone trust-region algorithm for nonlinear optimization subject to convex constraints. Math. Progr. 77, 69–94 (1997)

43.

Ulbrich, M., Ulbrich, S.: Nonmonotone trust region methods for nonlinear equality constrained optimization without a penalty function. Math. Progr. 95, 103–135 (2003)MathSciNetCrossRefMATH

44.

Xiao, Y., Zhou, F.J.: Nonmonotone trust region methods with curvilinear path in unconstrained optimization. Computing 48, 303–317 (1992)MathSciNetCrossRefMATH

45.

Xiao, Y., Chu, E.K.W.: Nonmonotone trust region methods, Technical Report 95/17. Monash University, Clayton, Australia (1995)

46.

Yuan, Y.: Conditions for convergence of trust region algorithms for nonsmooth optimization. Math. Progr. 31, 220–228 (1985)CrossRefMATH

47.

Zhang, H.C., Hager, W.W.: A nonmonotone line search technique for unconstrained optimization. SIAM J. Optim. 14(4), 1043–1056 (2004)MathSciNetCrossRefMATH

48.

Zhou, F., Xiao, Y.: A class of nonmonotone stabilization trust region methods. Computing 53(2), 119–136 (1994)MathSciNetCrossRefMATH

Title: Two globally convergent nonmonotone trust-region methods for unconstrained optimization
Authors: Masoud Ahookhosh
Susan Ghaderi
Publication date: 01-02-2016
Publisher: Springer Berlin Heidelberg
Published in: Journal of Applied Mathematics and Computing / Issue 1-2/2016
Print ISSN: 1598-5865
Electronic ISSN: 1865-2085
DOI: https://doi.org/10.1007/s12190-015-0883-9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1-2/2016

On Hermitian positive definite solutions of the nonlinear matrix equation

A new parallel subspace correction method for advection–diffusion equation

Solvability of a coupled system of nonlinear fractional differential equations with fractional integral conditions

An iterative technique for solving singularly perturbed parabolic PDE

Global asymptotical stability for a diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal delay

Seasonal influences on a prey–predator model

Premium Partner