An Improved Trust Region Method Equipped by Filter Technique

Document Type : Original Scientific Paper

Authors

1 Department of Applied Mathematics, University of Kashan, Kashan, I. R. Iran

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

Abstract

Using a filter technique, a new efficient nonmonotone trust region method is proposed. The proposed scheme is based on updating the approximation of the Hessian matrix with the scaled memoryless BFGS update formula. To update the trust region radius, an appropriate adaptive scheme is used. Moreover, a proper nonmonotone procedure is applied. Assuming some suitable assumptions, the global convergence is obtained. Numerical results are reported to show the efficiency of the offered approach.

Keywords


[1] M. Ahookhosh and K. Amini, An efficient nonmonotone trust-region method for unconstrained optimization, Numer. Algor. 59 (2011) 523 − 540.
[2] N. Andrei, Accelerated scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Eur. J. Oper. Res. 204 (2010) 410 − 420.
[3] F. Arzani and M. R. Peyghami, A new nonmonotone fillter Barzilai-Borwein method for solving unconstrained optimization problems, Int. J. Comput. Math. (2014). 
[4] D. Ataee Tarzanagh, Z. Saeedian, M. R. Peyghami and H. Mesgarani, A new trust region method for solving least-squares transformation of system of equalities and inequalities, Math. Program. Stud. 9 (2015) 283 − 310.
[5] J. Barzilai and J. M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal. 8 (1988) 141 − 148.
[6] R. M. Chamberlain, M. J. D. Powell, C. Lemarechal and H. C. Pedersen, The watchdog technique for forcing convergence in algorithm for constrained optimization, Math. Program. Stud. 16 (1982) 1 − 17.
[7] A. Conn, N. Gould and Ph. L.Toint, Trust Region Methods, MPSSIAM Series on Optimization, SIAM, Philadelphia, 2000.
[8] E. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program. 91 (2002) 201 − 213.
[9] J. Y. Fan and Y. X. Yuan, A new trust region algorithm with trust region radius converging to zero, Proc. 5th Inter. Conf. Optim. Tech. Appl., Hong Kong, 15-17 December 2001, 786 − 794.
[10] M. Fatemi and N. Mahdavi-Amiri, A filter trust-region algorithm for unconstrained optimization with strong global convergence properties, Comput. Optim. Appl. 52 (2012) 239 − 266.
[11] M. Fatemi and N. Mahdavi-Amiri, A non-monotone trust region algorithm for unconstrained optimization with dynamic reference iteration updates using filter, Optimization 61 (6) (2012) 733 − 763.
[12] R. Fletcher and S. Leyer, Nonlinear programming without a penalty function, Math. Program. Ser. A 91 (2002) 239 − 269.
[13] N. I. M. Gould, D. Orban, A. Sartenaer and PhL. Toint, Sensitivity of trust region algorithms to their parameters, Q. J. Oper. Res. 3 (2005) 227 − 241.
[14] N. I. M. Gould, D. Orban and P. L. Toint, Cuter and sifdec: a constrained and unconstrained testing environment, revisited, ACM Trans. Math. Softw. 29 (4) (2002) 373 − 394.
[15] L. Grippo, F. Lampariello and S. Lucidi, A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal. 23 (1986) 707 − 716.
[16] L. Grippo and M. Sciandrone, Nonmontone globalization techniques for the Barzilai-Borwein gradient method, Comput. Optim. Appl. 23 (2002) 143−169.
[17] A. Kamandi, K. Amini and M. Ahookhosh, An improved adaptive trust-region algorithm, Optim. Lett. 11 (3) (2017) 555 − 569.
[18] A. Kamandi and K. Amini, A new nonmonotone adaptive trust region algorithm, Appl. Math. 67 (2021) 233-250.
[19] L. Grippo, F. Lampariello and S. Lucidi, A truncated Newton method with nonmonotone line search for unconstrained optimization, J. Optim. Theory Appl. 60 (3) (1989) 401 − 419.
[20] D. Li and M. Fukushima, A modified BFGS method and its global convergence in nonconvex minimization, J. Comput. Appl. Math. 129 (2001) 1535.
[21] J. J. Moré, B. S. Garbow and K. E. Hilstron, Testing unconstrained optimization software, ACM Trans. Math. Softw. 7 (1981) 17 − 41.
[22] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, NewYork, 2007.
[23] J. Nocedal and Y. Yuan, Combining trust region and line search techniques, In: Y. Yuan (ed.), Advanced in Nonlinear Programming, Kluwer Academic, Dordrecht, (1996) 153 − 175.
[24] Z. Saeidian and F. Arzani (in Persian), Solving the Unconstrained Optimization Problems Using the Combination of Nonmonotone Trust Region Algorithm and Filter Technique, J. Oper. Res. Appl. 17 (1) (2020) 85 − 101.
[25] Z. Saeidian and M. R. Peyghami, An adaptive nonmonotone trust region method for unconstrained optimization problems based on a simple subproblem, Iran. J. Numer. Anal. Optim. 5 (2) (2015) 95 − 117.
[26] Z. Sang and Q. Sun, A self-adaptive trust region method with line search based on a simple subproblem model, J. Appl. Math. Comput. 232 (2) (2009) 514 − 522.
[27] A. Sartenaer, Automatic determination of an initial trust region in nonlinear programming, SIAM J. Sci. Comput. 18 (6) (1997) 1788 − 1803.
[28] Z. Shi and J. Guo, A new trust region method for unconstrained optimization, J. Comput. Appl. Math. 213 (2008) 509 − 520.
[29] Z. J. Shi and H. Q. Wang, A new self-adaptive trust region method for unconstrained optimization, Technical Report, College of Operations Research and Management, Qufu Normal University, 2004.
[30] Y. Xue, H. Liu and Z. Liu, An improved nonmonotone adaptive trust region method, Appl. Math. 64 (3) (2019) 335 − 350.
[31] H. C. Zhang and W. W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004) 1043 − 1056.
[32] X. S. Zhang, J. L. Zhang and L. Z. Liao, An adaptive trust region method and its convergence, Sci. China. 45 (2002) 620 − 631.