A nonmonotone ODE-based method for unconstrained optimization

This paper presents a new hybrid algorithm for unconstrained optimization problems, which combines the idea of the IMPBOT algorithm with the nonmonotone line search technique. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step, via a modified limited-memory BFGS two loop recursion that requires only matrix–vector products, thus reducing the computations and storage. Furthermore, when the trial step is not accepted, the proposed method performs a line search along it using a modified nonmonotone scheme, thus a larger stepsize can be yielded in each line search procedure. Under some reasonable assumptions, the convergence properties of the proposed algorithm are analysed. Numerical results are also reported to show the efficiency of this proposed method.

[1]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[2]  L. Grippo,et al.  A nonmonotone line search technique for Newton's method , 1986 .

[3]  E. Michael Gertz,et al.  A quasi-Newton trust-region method , 2004, Math. Program..

[4]  Ping Hu,et al.  A Nonmonotone Line Search Slackness Technique for Unconstrained Optimization , 2013, J. Optim. Theory Appl..

[5]  J. Borwein,et al.  Two-Point Step Size Gradient Methods , 1988 .

[6]  María C. Maciel,et al.  Monotone and nonmonotone trust-region-based algorithms for large scale unconstrained optimization problems , 2013, Comput. Optim. Appl..

[7]  Desmond J. Higham,et al.  Trust Region Algorithms and Timestep Selection , 1999, SIAM J. Numer. Anal..

[8]  Yuhong Dai.  On the Nonmonotone Line Search , 2002 .

[9]  Shengquan Wang,et al.  Nonmonotone adaptive trust region method , 2011, Eur. J. Oper. Res..

[10]  Zhiwei Xu,et al.  The convergence of subspace trust region methods , 2009, J. Comput. Appl. Math..

[11]  Yigui Ou.  An ODE-Based Trust Region Filter Algorithm for Unconstrained Optimization , 2011 .

[12]  Jorge Nocedal,et al.  Combining Trust Region and Line Search Techniques , 1998 .

[13]  Liqun Qi,et al.  Neurodynamical Optimization , 2004, J. Glob. Optim..

[14]  N. Deng,et al.  Nonmonotonic trust region algorithm , 1993 .

[15]  Jorge J. Moré,et al.  Benchmarking optimization software with performance profiles , 2001, Math. Program..

[16]  Li-Zhi Liao,et al.  A continuous Newton-type method for unconstrained optimization , 2007 .

[17]  M. Bartholomew-Biggs,et al.  Some effective methods for unconstrained optimization based on the solution of systems of ordinary differential equations , 1989 .

[18]  Marcos Raydan,et al.  The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem , 1997, SIAM J. Optim..

[19]  Zheng-Hai Huang,et al.  A Non-monotone Line Search Algorithm for Unconstrained Optimization , 2010, J. Sci. Comput..

[20]  Masoud Ahookhosh,et al.  A nonmonotone trust-region line search method for large-scale unconstrained optimization , 2012 .

[21]  Z. J. Shi,et al.  New Inexact Line Search Method for Unconstrained Optimization , 2005 .

[22]  Neng-zhu Gu,et al.  Incorporating nonmonotone strategies into the trust region method for unconstrained optimization , 2008, Comput. Math. Appl..

[23]  L. Liao,et al.  Combining Trust-Region Techniques and Rosenbrock Methods to Compute Stationary Points , 2009 .

[24]  Yijun Li,et al.  A new nonmonotone trust-region method of conic model for solving unconstrained optimization , 2010, J. Comput. Appl. Math..

[25]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[26]  Philippe L. Toint,et al.  An Assessment of Nonmonotone Linesearch Techniques for Unconstrained Optimization , 1996, SIAM J. Sci. Comput..

[27]  Yigui Ou,et al.  A hybrid trust region algorithm for unconstrained optimization , 2011 .

[28]  Qing-jun Wu,et al.  Nonmonotone trust region algorithm for unconstrained optimization problems , 2010, Appl. Math. Comput..

[29]  Jorge Nocedal,et al.  Representations of quasi-Newton matrices and their use in limited memory methods , 1994, Math. Program..

[30]  Ya-Xiang Yuan,et al.  Optimization Theory and Methods: Nonlinear Programming , 2010 .

[31]  Neculai Andrei,et al.  An Unconstrained Optimization Test Functions Collection , 2008 .

[32]  Yi-gui Ou,et al.  A hybrid ODE-based method for unconstrained optimization problems , 2012, Comput. Optim. Appl..

[33]  William W. Hager,et al.  A Nonmonotone Line Search Technique and Its Application to Unconstrained Optimization , 2004, SIAM J. Optim..

[34]  Masoud Ahookhosh,et al.  Optimization: a Journal of Mathematical Programming and Operations Research a Class of Nonmonotone Armijo-type Line Search Method for Unconstrained Optimization a Class of Nonmonotone Armijo-type Line Search Method for Unconstrained Optimization , 2022 .

[35]  Wu Qing-jun.  Nonmonotone trust region algorithm for unconstrained optimization problems , 2010 .

[36]  Wenyu Sun,et al.  Nonmonotone trust region method for solving optimization problems , 2004, Appl. Math. Comput..