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A nonmonotone trust region method with new inexact line search for unconstrained optimization

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Abstract

In this paper, a new nonmonotone inexact line search rule is proposed and applied to the trust region method for unconstrained optimization problems. In our line search rule, the current nonmonotone term is a convex combination of the previous nonmonotone term and the current objective function value, instead of the current objective function value . We can obtain a larger stepsize in each line search procedure and possess nonmonotonicity when incorporating the nonmonotone term into the trust region method. Unlike the traditional trust region method, the algorithm avoids resolving the subproblem if a trial step is not accepted. Under suitable conditions, global convergence is established. Numerical results show that the new method is effective for solving unconstrained optimization problems.

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References

  1. Ahookhosh, M., Amini, K.: A nonmotone trust region method with adaptive radius for unconstrained optimization. Comput. Math. Appl. 60, 411–422 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ahookhosh, M., Amini, K., Peyghami, M.R.: A nonmonotone trust-region line search method for large-scale unconstrained optimization. Appl. Math. Model. 36, 478–487 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Touati-Ahmed, D., Storey, C.: Efficient hybrid conjugate gradient techniques. J. Optim. Theory Appl. 64(2), 379–397 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  4. Conn, A.R., Gould, N.I.M., Toint, Ph.L.: Trust-region Methods. SIAM Publications, Philadelphia, PA (2000)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Deng, N.Y., Xiao, Y., Zhou, F.J.: Nonmonotonic trust region algorithm. J. Optim. Theory Appl. 76(2), 259–285 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Dai, Y.H.: A nonmonotone conjugate gradient algorithm for unconstrained optimization. J. Syst. Sci. Complex 15(2), 139–145 (2002)

    MathSciNet  MATH  Google Scholar 

  9. Fletcher, R.: An algorithm for solving linearly constrained optimization problems. Math. Program. 2, 133–165 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fu, J., Sun, W.: Nonmotone adaptive trust region method for unconstrained optimization problems. Appl. Math. Comput. 163(1), 489–504 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gertz, E.M.: Combination trust-region line search methods for unconstrained optimization. Ph.D. thesis, University of California, San Diego (1999)

  12. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23(4), 707–716 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gu, N.Z., Mo, J.T.: Incorporating nonmonotone strategies into the trust region method for unconstrained optimization. Appl. Math. Comput. 55, 2158–2172 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Grippo, L., Sciandrone, M.: Nonmonotone globalization techniques for the Barzilai-Borwein gradient method. Comput. Optim. Appl. 23, 143–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Ke, X., Han, J.: A class of nonmotone trust region algorithms for unconstrained optimization. Sci. China Ser. A 41(9), 927–932 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liu, G., Han, J., Sun, D.: Global convergence of BFGS algorithm with nonmonotone linesearch. Optim. 34, 147–159 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mo, J.T., Liu, C.Y., Yan, S.C.: A nonmonotone trust region method based on nonincreasing technique of weighted average of the successive function values. J. Comput. Appl. Math. 209, 97–108 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. More, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math. Softw. 7(1), 17–41 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nocedal, J., Yuan, Y.: Combining trust region and line search techniques. In: Yuan, Y. (ed.) Advances in Nonlinear Programming, pp. 153–175. Kluwer Academic Publishers, Dordrecht (1996)

    Google Scholar 

  20. Powell, M.J.D.: Convergence properties of a class minimization algorithms. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds.) Nonlinear Programming, vol. 2, pp. 1–25. Academic Press, New York (1975)

    Google Scholar 

  21. Powell, M.J.D., Yuan, Y.: A trust region algorithm for equality constrained optimization. Math. Program. 49, 189–213 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, W.Y.: Nonmonotone trust region method for solving optimization problems. Appl. Math. Comput. 156(1), 59–174 (2004)

    Article  MathSciNet  Google Scholar 

  23. Shi, Z.J., Shen, J.: New inexact line search method for unconstrained optimization. J. Optim. Theory Appl. 127(2), 425–446 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shi, Z.J., Sun, G.: A diagonal-sparse quasi-Newton method for unconstrained optimization problem. J. Sys. Sci Math. Sci. 26(1), 101–112 (2006)

    MathSciNet  MATH  Google Scholar 

  25. Sun, W.Y., Han, J.Y., Sun, J.: On the global convergence of nonmonotone descent methods. J. Comput. Appl. Math. 146, 89–98 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shi, Z.J., Wang, S.Q.: Nonmonotone adaptive trust region method. Eur. J. Oper. Res. 208, 28–36 (2011)

    Article  MATH  Google Scholar 

  27. Toint, Ph.L.: An assessment of non-monotone line search techniques for unconstrained optimization. SIAM J. Sci. Stat. Comput. 17(3), 725–739 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yuan, Y.: On a subproblem of trust region algorithms for constrained optimization. Math. Program. 47, 53–63 (1990)

    Article  MATH  Google Scholar 

  29. Zhang, H., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14(4), 1043–1056 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350007, People’s Republic of China

    Jinghui Liu & Changfeng Ma

Authors
  1. Jinghui Liu
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  2. Changfeng Ma
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Correspondence to Changfeng Ma.

Additional information

The project is supported by National Natural Science Foundation of China (Grant No.11071041) and Fujian Natural Science Foundation (Grant No.2009J01002).

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Liu, J., Ma, C. A nonmonotone trust region method with new inexact line search for unconstrained optimization. Numer Algor 64, 1–20 (2013). https://doi.org/10.1007/s11075-012-9652-0

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