nach oben

Calcolo

Erschienen in:

01.09.2017

A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints

verfasst von: Morteza Kimiaei

Erschienen in: Calcolo | Ausgabe 3/2017

Einloggen, um Zugang zu erhalten

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

A nonmonotone trust-region method for the solution of nonlinear systems of equations with box constraints is considered. The method differs from existing trust-region methods both in using a new nonmonotonicity strategy in order to accept the current step and a new updating technique for the trust-region-radius. The overall method is shown to be globally convergent. Moreover, when combined with suitable Newton-type search directions, the method preserves the local fast convergence. Numerical results indicate that the new approach is more effective than existing trust-region algorithms.

Vorheriger Artikel Backward difference formulae for Kuramoto–Sivashinsky type equations

Nächster Artikel Quarkonial frames with compression properties

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

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

Ahookhosh, M., Amini, K., Kimiaei, M.: A globally convergent trust-region method for large-scale symmetric nonlinear systems. Numer. Funct. Anal. Optim. 36, 830–855 (2015)MathSciNetCrossRefMATH

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

Bellavia, S., Macconi, M., Morini, B.: An affine scaling trust-region approach to bound-constrained nonlinear systems. Appl. Numer. Math. 44, 257–280 (2003)MathSciNetCrossRefMATH

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

Bellavia, S., Morini, B.: An interior global method for nonlinear systems with simple bounds. Optim. Methods Softw. 20, 1–22 (2005)MathSciNetCrossRefMATH

Bellavia, S., Morini, B.: Subspace trust-region methods for large bound-constrained nonlinear equations. SIAM J. Numer. Anal. 44, 1535–1555 (2006)MathSciNetCrossRefMATH

Bellavia, S., Morini, B., Pieraccini, S.: Constrained dogleg methods for nonlinear systems with simple bounds. Comput. Optim. Appl. 53, 771–794 (2011)MathSciNetCrossRefMATH

10.

Bellavia, S., Pieraccini, S.: On affine scaling inexact dogleg methods for bound-constrained nonlinear systems. Optim. Methods Softw. 30, 276–300 (2015)MathSciNetCrossRefMATH

11.

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

12.

Dai, Y.H.: On the nonmonotone line search. J. Optim. Theory Appl. 112, 315–330 (2002)MathSciNetCrossRefMATH

13.

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

14.

Dennis, J.E., Vicente, L.N.: Trust-region interior-point algorithms for minimization problems with simple bounds. In: Fischer, H., Riedmüller, B., Schäffler, S. (eds.) Applied Mathematics and Parallel Computing, pp. 97–107. Festschrift for Klaus Ritter. Physica, Heidelberg (1996)CrossRef

15.

Dirkse, S.P., Ferris, M.C.: MCPLIB: a collection of nonlinear mixed complementary problems. Optim. Methods Softw. 5, 319–345 (1995)CrossRef

16.

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

17.

Esmaeili, H., Kimiaei, M.: A new adaptive trust-region method for system of nonlinear equations. Appl. Math. Model. 38, 3003–3015 (2014)MathSciNetCrossRef

18.

Esmaeili, H., Kimiaei, M.: An efficient adaptive trust-region method for systems of nonlinear equations. Int. J. Comput. Math. 92, 151–166 (2015)MathSciNetCrossRefMATH

19.

Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM J. Optim. 9, 14–32 (1999)MathSciNetCrossRefMATH

20.

Facchinei, F., Pang, J.-S.: Finite-dimensional variational inequalities and complementarity problems, volume i. Springer Series in Operations Research, New York (2003)MATH

21.

Fan, J.Y., Pan, J.Y.: A modified trust region algorithm for nonlinear equations with new updating rule of trust region radius. Int. J. Comput. Math. 87, 3186–3195 (2010)MathSciNetCrossRefMATH

22.

Floudas, C.A., Pardalos, P.M., et al.: Handbook of Test Problems in Local and Global Optimization. Nonconvex Optimization and its Applications 33. Kluwer Academic Publishers, Germany (1999)

23.

Gould, N.I.M., Orban, D., Sartenaer, A., Toint, PhL: Sensitivity of trust-region algorithms to their parameters. 4OR Q. J. Oper. Res. 3, 227–241 (2005)MathSciNetCrossRefMATH

24.

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

25.

Grippo, L., Lampariello, F., Lucidi, S.: A class of nonmonotone stabilization methods in unconstrained optimization. Numerische Mathematik 59, 779–805 (1991)MathSciNetCrossRefMATH

26.

Jiang, H., Fukushima, M., Qi, L., Sun, D.: A trust-region method for solving generalized complementarity problems. SIAM J. Optim. 8, 140–157 (1998)MathSciNetCrossRefMATH

27.

Kanzow, C.: An active set-type Newton method for constrained nonlinear systems. In: Ferris, M.C., Mangasarian, O.L., Pang, J.-S. (eds.) Complementarity: Applications, Algorithms and Extensions, pp. 179–200. Kluwer Academic Publishers, Dordrecht (2001)CrossRef

28.

Kanzow, C., Klug, A.: On affine-scaling interior-point Newton methods for nonlinear minimization with bound constraints. Comput. Optim. Appl. 35, 177–197 (2006)MathSciNetCrossRefMATH

29.

Kanzow, C., Klug, A.: An interior-point affine-scaling trust-region method for semismooth equations with box constraints. Comput. Optim. Appl. 37, 329–353 (2007)MathSciNetCrossRefMATH

30.

Kanzow, C., Qi, H.: A QP-free constrained Newton-type method for variational inequality problems. Math. Program. 85, 81–106 (1999)MathSciNetCrossRefMATH

31.

Kanzow, C., Zupke, M.: Inexact trust-region methods for nonlinear complementarity problems. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth Semismooth and Smoothing Methods, pp. 211–233. Kluwer Academic Publishers, Dordrecht, Netherlands (1999)

32.

Lukšan, L., Vlček, J.: Sparse and partially separable test problems for unconstrained and equality constrained optimization. Institute of Computer Science, Academy of Sciences in the Czech Republic, Technical Report, No. 767 (1999)

33.

Meintjes, K., Morgan, A.P.: Chemical equilibrium systems as numerical tests problems. ACM Trans. Math. Softw. 16, 143–151 (1990)CrossRefMATH

34.

Moré, J.J., Garbow, B.S., Hillström, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7, 17–41 (1981)MathSciNetCrossRefMATH

35.

Morini, B., Porcelli, M.: TRESNEI: a Matlab trust-region solver for systems of nonlinear equalities and inequalities. Comput. Optim. Appl. 51, 27–49 (2012)MathSciNetCrossRefMATH

36.

Powell, M.J.D.: Convergence properties of a class minimization algorithm. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds.) Nonlinear Programming 2, pp. 1–27. Academic Press, New York (1975)

37.

Qi, L., Tong, X.J., Li, D.H.: Active-set projected trust-region algorithm for box-constrained nonsmooth equations. J. Optim. Theory Appl. 120, 601–625 (2004)MathSciNetCrossRefMATH

38.

Sartenaer, A.: Automatic determination of an initial trust region in nonlinear programming. SIAM J. Sci. Comput. 18, 1788–1803 (1997)MathSciNetCrossRefMATH

39.

Toint, PhL: An assessment of nonmonotone linesearch techniques for unconstrained optimization. SIAM J. Sci. Comput. 17, 725–739 (1996)MathSciNetCrossRefMATH

40.

Tsoulos, I.G., Stavrakoudis, A.: On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods. Nonlinear Anal. Real World Appl. 11, 2465–2471 (2010)MathSciNetCrossRefMATH

41.

Ulbrich, M.: Non-monotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. SIAM J. Optim. 11, 889–917 (2001)MathSciNetCrossRefMATH

42.

Ulbrich, M., Ulbrich, S., Heinkenschloss, M.: Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to pointwise constraints. SIAM J. Control Optim. 37, 731–764 (1999)MathSciNetCrossRefMATH

43.

Xu, L., Burke, J.V.: ASTRAL: An active set \(\ell _{\infty }-\)trust-region algorithm for box constrained optimization. Technical Report, Department of Mathematics, University of Washington, Seattle (2007)

44.

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

45.

Zhang, J., Wang, Y.: A new trust region method for nonlinear equations. Math. Methods Oper. Res. 58, 283–298 (2003)MathSciNetCrossRefMATH

46.

Zhu, D.: Affine scaling interior Levenberg-Marquardt method for bound-constrained semismooth equations under local error bound conditions. J. Comput. Appl. Math. 219, 198–215 (2008)MathSciNetCrossRefMATH

Titel: A new class of nonmonotone adaptive trust-region methods for nonlinear equations with box constraints
verfasst von: Morteza Kimiaei
Publikationsdatum: 01.09.2017
Verlag: Springer Milan
Erschienen in: Calcolo / Ausgabe 3/2017
Print ISSN: 0008-0624
Elektronische ISSN: 1126-5434
DOI: https://doi.org/10.1007/s10092-016-0208-x

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Weitere Artikel der Ausgabe 3/2017

Quarkonial frames with compression properties

Numerical validation for systems of absolute value equations

A posteriori error analysis of an augmented mixed-primal formulation for the stationary Boussinesq model

Square roots of matrices

A parameter-uniform second order numerical method for a weakly coupled system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms

Discrete p-robust -liftings and a posteriori estimates for elliptic problems with source terms