An adaptive nonmonotone trust region method for unconstrained optimization problems based on a simple subproblem

Document Type : Research Article


1 Faculty of Mathematics, K.N. Toosi University of Technology, Tehran, Iran.

2 Scientific Computations in OPtimization and Systems Engineering (SCOPE), K.N. Toosi University of Technology, Tehran, Iran.


Using a simple quadratic model in the trust region subproblem, a new adaptive nonmonotone trust region method is proposed for solving unconstrained optimization problems. In our method, based on a slight modification of the proposed approach in (J. Optim. Theory Appl. 158(2):626-635, 2013), a new scalar approximation of the Hessian at the current point is provided. Our new proposed method is equipped with a new adaptive rule for updating the radius and an appropriate nonmonotone technique. Under some suitable and standard assumptions, the local and global convergence properties of the new algorithm as well as its convergence rate are investigated. Finally, the practical performance of the new proposed algorithm is verified on some test problems and compared with some existing algorithms in the literature.


1. Ahookhosh, M. and Amini, K. A nonmonotone trust region method with adaptive radius for unconstrained optimization, Comput. Math. Appl. 60(2010) 411-422.
2. Ahookhosh, M. and Amini, K. An ecient nonmonotone trust-region method for unconstrained optimization, Numer. Algorithms 59(2011) 523-540.
3. Ahookhosh, M., Amini, K. and Peyghami, M.R. A nonmonotone trustregion line search method for large- scale unconstrained optimization, Appl. Math. Model. 36(2012) 478-487.
4. Andrei, N. An unconstrained optimization test functions collection, Adv. Model. Optim. 10(1)(2008) 147-161.
5. Ataee Tarzanagh, D., Peyghami, M. R. and Mesgarani, H. A new nonmonotone trust region method for unconstrained optimization equipped by an efficient adaptive radius, Optim. Methods Softw. 29(4)(2014) 819-836.
6. Biglari, F., Hassan, M. and Leong, W. J. New quasi-Newton methods via higher order tensor models, J. Comput. Appl. Math. 235(2011) 2412-2422.
7. Biglari, F. and Solimanpur, M. Scaling on the Spectral Gradient Method, J. Optim. Theory Appl. 158(2)(2013) 626-635.
8. Chamberlain, R. M., Powell, M. J. D., Lemarechal, C. and Pedersen, H. C. The watchdog technique for forcing convergence in algorithm for constrained optimization, Math. Program. Stud. 16(1982) 1-17.
9. Conn, A., Gould, N. and Toint, Ph. L. Trust Region Methods, SIAM, Philadelphia, 2000.
10. Dolan, E. and More, J. J. Benchmarking optimization software with performance profiles, Math. Program. 91(2002) 201-213.
11. Fan, J. Y. and Yuan, Y. X. A new trust region algorithm with trust region radius converging to zero, Proceedings of the 5th International Conference on Optimization, Techniques and Applications, 2001.
12. Gertz, E. M. A quasi-Newton trust-region method, Math. Program. Ser. A. 100(3)(2004) 447-470.
13. Grippo, L., Lampariello, F. and Lucidi, S. A nonmonotone line search technique for Newton's method, SIAM J. Numer. Anal. 23(1986) 707-716.
14. Grippo, L., Lampariello, F. and Lucidi, S. A truncated Newton method with nonmonotone line-search for unconstrained optimization, J. Optim. Theory Appl. 60(1989) 401-419.
15. Li, G. D. A trust region method with automatic determination of the trust region radius, Chinese J. Engry. Math. 23(2006) 843-848.
16. Morie, J. J., Garbow, B. S. and Hilstron, K. E. Testing unconstrained optimization software, ACM Tran. Math. Software 7(1981) 17-41.
17. Nocedal, J. and Wright, S. J. Numerical Optimization, Springer, New York, 2006.
18. Nocedal, J. and Yuan, Y. Combining trust region and line search techniques, In: Y. Yuan (ed.), Advanced in Nonlinear Programming, Kluwer Academic, Dordrecht, 153-175, 1996.
19. Powell, M. J. D. On the global convergence of trust region algorithms for unconstrained optimization, Math. Program. 29(1984) 297-303.
20. Sang, Z. and Sun, Q. A self-adaptive trust region method with line search based on a simple subproblem model, J. Appl. Math. Comput. 232(2009)514-522.
21. Sartenaer, A. Automatic determination of an initial trust region in nonlinear programming, SIAM J. Sci. Comput. 18(6)(1997) 1788-1803.
22. Shi, Z. and Guo, J. A new trust region method for unconstrained optimization, J. Comput. Appl. Math. 213(2008) 509-520.
23. Shi, Z. J. and Wang, H. Q. A new self-adaptive trust region method for unconstrained optimization, Technical Report, College of Operations Research and Management, Qufu Normal University, 2004.
24. Shultz, G. A., Schnabel, R. B. and Byrd, A. R. H. A family of trustregion-based algorithm for unconstrained minimization with strong global convergence properties, SIAM J. Numer. Anal. 22(1)(1985) 47-67.
25. Toint, Ph. L. Non-monotone trust-region algorithm for nonlinear optimization subject to convex constraints, Math. Program. 77(1997) 69–94.
26. Wei, Z., Li, G. and Qi, L. New quasi-Newton methods for unconstrained optimization problems, Appl. Math. Comput. 175(2006) 1156–1188.
27. Zhang, H. C. and Hager, W. W. A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14(2004) 1043–1056.
28. Zhang, X. S., Zhang, J. L. and Liao, L. Z. A nonmonotoneadaptive
trust region method and its convergence, Int. J. Comput. Math. Appl. 45(2003) 1469–1477.
29. Zhang, X. S., Zhang, J. L. and Liao, L. Z. An adaptive trust region method and its convergence, Sci. China 45(2002) 620–631.
30. Zhou, Q. and Zhang, C. A new nonmonotone adaptive trust region method based on simple quadratic models, J. Appl. Math. Comput. 40(2012) 111–123.