A new dwindling nonmonotone filter method without gradient information for solving large-scale systems of equations

Iranian Journal of Numerical Analysis and Optimization

مقاله 2، دوره 8، شماره 1 - شماره پیاپی 13، خرداد 2018، صفحه 19-40   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.v8i1.49742

نویسندگان

F. Arzani؛ M.R. Peyghami^*

K.N. Toosi University of Technology,

چکیده

In this paper, we present a new derivative-free spectral residual method for solving large-scale systems of equations. Our algorithm is equipped with a dwindling multidimensional nonmonotone filter in which whose envelope is dwindling as the step-length of line search is decreasing. The proposed algorithm is also combined with a relaxed nonmonotone line search technique which allows the algorithm to enjoy the nonmonotone property from scratch. Under some standard assumptions, the global convergence property of the proposed algorithm is established. Numerical results on some test problems show the efficiency and effectiveness of the new algorithm in practice.

کلیدواژه‌ها

Dwindling filter technique؛ Systems of equations؛ Nonmonotone line search؛ Global convergence

مراجع

1. Ahookhosh, M. and Amini, K. A nonmonotone trust region method with adaptive radius for unconstrained optimization, Comput. Math. Appl. 60(2010), no. 3, 411–422.

2. Ahookhosh, M., Amini, K., and Reza Peyghami, M. A nonmonotone trust region line search method for large-scale unconstrained optimization, Appl.Math. Model. 36 (2012), no. 1, 478–487.

3. Andrei, N. An unconstrained optimization test functions collection, Adv. Model. Optim. 10 (2008), no. 1, 147–161.

4. Arzani, F. and Reza Peyghami, M. An approach based on dwindling filter method for positive definite generalized eigenvalue problem, Comp. Appl. Math. (2016). doi:10.1007/s40314-016-0391-z

5. Arzani, F. and Reza Peyghami, M. A new nonmonotone filter Barzilai Borwein method for solving unconstrained optimization problems, Int. J. Comput. Math. 93 (2016), no. 3, 596–608.

6. Ataee Tarzanagh, D., Saeidian, Z., Reza Peyghami, M., and Mesgarani, H. A new trust region method for solving least-square transformation of system of equalities and inequalities, Optim. Lett. 9 (2015), no. 2, 283–310.

7. Barzilai, J. and Borwein, J. M. Two-point step size gradient methods, IMA J. Numer. Anal. 8 (1988), 141–148.

8. Chen, Y. and Sun, W. A dwindling filter line search method for unconstrained optimization, Math. Comp. 84 (2015), no. 291, 187–208.

9. Cheng, W. Y. A two-term PRP based-descent method, Numer. Funct. Anal. Optim. 28 (2007), no. 11-12, 1217–1230.

10. Cheng, W. and Chen, Z. Nonmonotone spectral method for large-scale symmetric nonlinear equations, Numer. Algorithms 62 (2013), 149–162.

11. Cheng, W. and Li, D. H. A derivative-free nonmonotone line search and its application to the spectral residual method, IMA J. Numer. Anal. 29 (2009), no. 3, 814–825.

12. Dolan, E. and Mor´e, J. J. Benchmarking optimization software with performance profiles, Math. Program. 91 (2002), no. 2, Ser. A, 201–213.

13. Fatemi, M. and Mahdavi-Amiri, N. A filter trust-region algorithm for unconstrained optimization with strong global convergence properties, Comput. Optim. Appl. 52 (2012), no. 1, 239–266.

14. Fatemi, M. and Mahdavi-Amiri, N. A non-monotone trust region algorithm for unconstrained optimization with dynamic reference iteration updates using filter, Optimization 61 (2012), no. 6, 733–763.

15. Fletcher, R. and Leyffer, S. Nonlinear programming without a penalty function, Math. Program. 91 (2002), no. 2, Ser. A, 239–269.

16. Fletcher, R., Leyffer, S., and Toint, Ph.L. A brief history of filter methods, SIAG/OPT Views-and-News, 18 (2006), 2–12.

17. Grippo, L., Lampariello, F., and Lucidi, S. A class of nonmonotone stabi lization methods in unconstrained optimization, Numer. Math. 59 (1991), no. 8, 779–805.

18. Grippo, L. and Sciandrone, M. Nonmonotone globalization techniques for the Barzilai-Borwein gradient method, Comput. Optim. Appl. 23 (2002), no. 2, 143–169.

19. Hock, W. and Schittkowski, K. Test examples for nonlinear program ming codes, Lecture Notes in Economics and Mathematical Systems, 187. Springer-Verlag, Berlin-New York, 1981.

20. Kanzow, C., Yamashita, N., and Fukushima, M. Levenberg–Marquardt methods for constrained nonlinear equations with strong local convergence properties, J. Comput. Appl. Math. 172 (2004), no. 2, 375–397.

21. La Cruz, W. A projected derivative-free algorithm for nonlinear equations with convex constraints, Optim. Methods Softw. 29 (2014), no. 1, 24–41.

22. La Cruz, W. Residual spectral algorithm for solving monotone equations on a Hilbert space, Appl. Math. Comput. 219 (2013), no. 12, 6633–6644.

23. La Cruz, W. A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic com putation, Appl. Math. Comput. 217 (2010), no. 1, 11–24.

24. La Cruz, W., Mart´ınez, J.M., and Raydan, M. Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comp. 75 (2006), no. 255, 1429–1448.

25. La Cruz, W. and Raydan, M. Nonmonotone spectral methods for large scale nonlinear systems, Optim. Methods Softw. 18 (2003), no. 5, 583–599.

26. Li, D. H. and Fukushima, M. A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optim. Methods Softw. 13 (2000), no. 3, 181–201.

27. Li, C. and Sun, W. On filter-successive linearization methods for nonlinear semidefinite programming, Sci. China Ser. A 52 (2009), no. 11, 2341–2361.

28. Peyghami, M.R. and Ataee Tarzanagh, D. A relaxed nonmonotone adaptive trust region method for solving unconstrained optimization problems, Comput. Optim. Appl. 61 (2015), no. 2, 321–341.

29. Sun, W. and Yuan, Y. Optimization Theory and Methods, Nonlinear programming. Springer Optimization and Its Applications, 1. Springer, New York, 2006.

30. Zhang, H. and Hager, W. W. A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14 (2004), no. 4, 1043–1056.

31. Zhang, Y., Sun, W., and Qi, L. A nonmonotone filter Barzilai-Borwein method for optimization, Asia-Pac. J. Oper. Res. 27 (2010), no. 1, 55–69.