Incorporating non-monotone trust region algorithm with line search method for unconstrained optimization

Document Type : Research Article

Authors

Department of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran

Abstract

This paper concerns an efficient trust region framework that exploits a new non-monotone line search method. The new algorithm avoids the sudden increase of the objective function values in the non-monotone trust region method. Instead of resolving the trust region subproblem whenever the trial step is rejected, the proposed algorithm employs an Armijo-type line search method in the direction of the rejected trial step to construct a new point. Global and superlinear properties are preserved under appropriate conditions. Comparative numerical experiments depict the efficiency and robustness of the new algorithm using the Dolan-More performance profiles.

Keywords

Main Subjects


[1] M. Ahookhosh, K. Amini, A non-monotone trust region method with adaptive radius for uncon-
strained optimization, Comput. Math. Appl. 60 (2010) 411–422.
[2] M. Ahookhoosh, K. Amini, M.R. Peyghami, A non-monotone trust region lin search method for
large scale unconstrained optimization, Appl. Math. Model. 36 (2012) 478–487.
[3] N. Andrei, An unconstrained optimization test functions collection, Adv. Model. Optim. 10 (2008)
147–161.
[4] L. Armijo, Minimization of functions having Lipschitz continuous first partial derivatives, Pac. J.
Math. 16 (1966) 1–3.
[5] R.M. Chamberlain, M.J.D. Powell, C. Lemarechal, H.C. Pedersen, The watchdog technique for
forcing convergence in algorithm for constrained optimization, Math. Program. Stud. 16 (1982)
1–17.
[6] A.R. Conn, N.I.M. Gould, P.L. Toint, Trust Region Methods, MPS/SIAM Series on Optimization,
SIAM, Philadelphia, 2000.
[7] N.Y. Deng, Y. Xiao, F.J. Zhou,Non-monotonic trust region algorithm, J. Optim. Theory. Appl. 76
(1993) 259–285.
[8] X. Ding, Q. Qu, X. Wang, A modified filter non-monotone adaptive retrospective trust region
method, PLOS ONE. 16 (2021) e0253016.
[9] E.D. Dolan, J.J. More, Benchmarking optimization software with performance profiles, Math. Pro-
gram. 91 (2002) 201–213.
[10] J.H. Fu, W.Y. Sun, Non-monotone adaptive trust region method for unconstrained optimization
problems, Appl. Math. Comput. 163 (2005) 489–504.
[11] E.M. Gertz, Combination Trust Region Line Search Methods for Unconstrained Optimization, Uni-
versity of California, San Diego, 1999.
[12] A.A. Goldstein, On steepest descent, SIAM J. Contro. 3 (1965) 147–151.
[13] L. Grippo, F. Lampariello, S. Lucidi, A non-monotone line search technique for Newtons method,
SIAM J. Numer. Anal. 23 (1986) 707–716.
[14] L. Grippo, F. Lampariello, S. Lucidi, A truncated Newton method with non-monotone line search
for unconstrained optimization, J. Optim. Theory. Appl. 60 (1989) 401–419.
[15] N.Z. Gu, J.T. Mo, Incorporating non-monotone strategies into the trust region method for uncon-
strained optimization, Appl. Math. Comput. 55 (2008) 2158–2172.
[16] A. Kamandi, K. Amini, A new non-monotone adaptive trust region algorithm, Appl. Math. 67
(2021) 233–250.
[17] J. Liu, C. Ma, A non-monotone trust region method with new inexact line search for unconstrained
optimization, Numer. Algorithms 64 (2013) 1–20.
[18] J. Nocedal, S.J. Wright, Numerical Optimization, Springer, NewYork, 2006.
[19] J. Nocedal, Y.X. Yuan, Combining trust region and line search techniques, In: Yuan, Y. (ed.) Ad-
vances in Nonlinear Programming, pp. 153–175. Kluwer Academic Publishers, Dordrecht, 1998.
[20] S. Rezaee, S. Babaie-Kafaki, A modified non-monotone trust region line search method, J. Appl.
Math. Comput. 57 (2018) 421–436.
[21] T. Steihaug, The conjugate gradient method and trust regions in large scale optimization, SIAM J.
Numer. Anal. 20 (1983) 626–637.
[22] P.L. Toint, An assessment of non-monotone line search techniques for unconstrained optimization,
SIAM J. Sci. Comput. 17 (1996) 725–739.
[23] Z. Wan, S. Huang, X.D. Zheng, New cautious BFGS algorithm based on modified Armijotype line
search, J. Inequal. Appl. 2012 (2012) 241.
[24] P. Wolfe, Convergence conditions for ascent methods, SIAM. Rev. 11 (1986) 226–235.
[25] X. Xiao, E.K.W. Chu, Non-monotone trust region methods, Technical Report 95/17. Monash Uni-
versity. Clayton, Australia, 1995.
[26] H. Zhang, W.W. Hager, A non-monotone line search technique and its application to unconstrained
optimization, SIAM J. Optim. 14 (2004) 1043–1056.
[27] F. Zhou, Y. Xiao, A class of non-monotone stabilization trust region methods, Computing 53 (1994)
119–136.