A Non-Monotone Trust Region Method with Non-Monotone Wolfe-Type Line Search Strategy for Unconstrained Optimization
Abstract: In this paper, we propose and analyze a non-monotone trust region method with non-monotone line search strategy for unconstrained optimization problems. Unlike the traditional non-monotone trust region method, our algorithm utilizes non-monotone Wolfe line search to get the next point if a trial step is not adopted. Thus, it can reduce the number of solving sub-problems. Theoretical analysis shows that the new proposed method has a global convergence under some mild conditions.
Cite this paper: Li, C. , Zhou, Q. and Wu, X. (2015) A Non-Monotone Trust Region Method with Non-Monotone Wolfe-Type Line Search Strategy for Unconstrained Optimization. Journal of Applied Mathematics and Physics, 3, 707-712. doi: 10.4236/jamp.2015.36084.
References

[1]   Nocedal, J. and Yuan, Y.X. (1998) Combining Trust Region and Line Search Techniques. In: Yuan, Y., Ed., Advanced in Nonlinear Programming, Kluwer Academic Publishers, Dordrecht, 153-175.
http://dx.doi.org/10.1007/978-1-4613-3335-7_7

[2]   Michael Gertz, E. (2004) A Quasi-Newton Trust Region Method. Mathematical Programming, 100, 447-470. http://dx.doi.org/10.1007/s10107-004-0511-1

[3]   Chamberlain, R.M. and Powell, M.J.D. (1982) The Watchdog Technique for Forcing Convergence in Algorithm for Constrained Optimization. Mathematical Programming Study, 16, 1-17.
http://dx.doi.org/10.1007/BFb0120945

[4]   Grippo, L., Lampariello, F. and Lucidi, S. (1986) A Nonmonotone Line Search Technique for Newton’s Method. Society for Industrial and Applied Mathematics, 23, 707-716.

[5]   Deng, N.Y., Xiao, Y. and Zhou, F.J. (1993) Nonmonotone Trust Region Algorithm. Journal of Optimization Theory and Application, 76, 259-285. http://dx.doi.org/10.1007/BF00939608

[6]   Toint, Ph.L. (1996) An Assessment of Nonmonotone Linesearch Technique for Unconstrained Optimization. Society for Industrial and Applied Mathematics, 17, 725-739.

[7]   Toint, Ph.L. (1997) Non-Monotone Trust-Region Algorithm for Nonlinear Optimization Subject to Convex Constraints. Mathmatical Programming, 77, 69-94. http://dx.doi.org/10.1007/BF02614518

[8]   Sun, W.Y. (2004) Nonmonotone Trust Region Method for Solving Optimization Problems. Applied Mathematics and Computation, 156, 159-174. http://dx.doi.org/10.1016/j.amc.2003.07.008

[9]   Mo, J.T., Zhang, K.C. and Wei, Z.X. (2005) A Nonmonotone Trust Region Method for Unconstrained Optimization. Applied Mathematics and Computation, 171, 371-384.
http://dx.doi.org/10.1016/j.amc.2005.01.048

[10]   Mo, J.T., Liu, C.Y. and Yan, S.C. (2007) A Nonmonotone Trust Region Method Based on Nonincreasing Technique of Weighted Average of the Successive Function Values. Journal of Computational and Applied Mathematics, 209, 97-108. http://dx.doi.org/10.1016/j.cam.2006.10.070

[11]   Ahookhosh, M. and Amini, K. (2012) An Efficient Nonmonotone Trust-Region Method for Unconstrained Optimization. Numerical Algorithms, 59, 523-540. http://dx.doi.org/10.1007/s11075-011-9502-5

[12]   Ahookhosh, M., Amini, K. and Bahrami, S. (2012) A Class of Nonmonotone Armijo-Type Line Search Method for Unconstrained Optimization. Optimization, 61, 387-404.
http://dx.doi.org/10.1080/02331934.2011.641126

[13]   Zhou, Q.Y., Chen, J. and Xie, Z.W. (2014) A Nonmonotone Trust Region Method Based on Simple Quadratic Models. Journal of Computational and Applied Mathematics, 272, 107-115.
http://dx.doi.org/10.1016/j.cam.2014.04.026

[14]   Huang, S., Wan, Z. and Chen, X.H. (2015) A New Nonmonotone Line Search Technique for Unconstrained Optimization. Numerical Algorithms, 68, 671-689.
http://dx.doi.org/10.1007/s11075-014-9866-4

[15]   Zhou, Q.Y. and Hang, D. (2015) Nonmonotone Adaptive Trust Region Method with Line Search Based on New Diagonal Updating. Applied Numerical Mathematics, 91, 75-88.
http://dx.doi.org/10.1016/j.apnum.2014.12.009

[16]   Gu, N.Z. and Mo, J.T. (2008) Incorporating Nonmonotone Strategies into the Trust Region for Unconstrained Optimization. Computers and Mathematics with Applications, 55, 2158-2172.
http://dx.doi.org/10.1016/j.camwa.2007.08.038

[17]   Ahookhosh, M., Amini, K. and Peyghami, M.R. (2012) A Nonmonotone Trust-Region Line Search Method for Large-Scale Unconstrained Optimization. Applied Mathematical Modelling, 36, 478-487.
http://dx.doi.org/10.1016/j.apm.2011.07.021

[18]   Zhang, H.C. and Hager, W.W. (2004) A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization. SIAM Journal on Optimization, 14, 1043-1056.
http://dx.doi.org/10.1137/S1052623403428208

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