A new adaptive trust region algorithm for optimization problems

Abstract It is well known that trust region methods are very effective for optimization problems. In this article, a new adaptive trust region method is presented for solving unconstrained optimization problems. The proposed method combines a modified secant equation with the BFGS updated formula and an adaptive trust region radius, where the new trust region radius makes use of not only the function information but also the gradient information. Under suitable conditions, global convergence is proved, and we demonstrate the local superlinear convergence of the proposed method. The numerical results indicate that the proposed method is very efficient.

[1]  Maojun Zhang,et al.  A three-terms Polak-Ribière-Polyak conjugate gradient algorithm for large-scale nonlinear equations , 2015, J. Comput. Appl. Math..

[2]  Guoyin Li,et al.  New quasi-Newton methods for unconstrained optimization problems , 2006, Appl. Math. Comput..

[3]  Gonglin Yuan,et al.  The global convergence of a modified BFGS method for nonconvex functions , 2018, J. Comput. Appl. Math..

[4]  Qing-hua Zhou,et al.  An improved trust region method for unconstrained optimization , 2013 .

[5]  Masoud Ahookhosh,et al.  A hybrid of adjustable trust-region and nonmonotone algorithms for unconstrained optimization , 2014 .

[6]  Gonglin Yuan,et al.  A modified Polak–Ribière–Polyak conjugate gradient algorithm for large-scale optimization problems , 2014 .

[7]  Gonglin Yuan,et al.  A Modified Hestenes-Stiefel Conjugate Gradient Algorithm for Large-Scale Optimization , 2013 .

[8]  Zhenjun Shi,et al.  A new trust region method for unconstrained optimization , 2008 .

[9]  Yong Wang,et al.  A new trust region method for nonlinear equations , 2003, Math. Methods Oper. Res..

[10]  Li-Zhi Liao,et al.  An adaptive trust region method and its convergence , 2002 .

[11]  Xiwen Lu,et al.  A BFGS trust-region method for nonlinear equations , 2011, Computing.

[12]  Ya-Xiang Yuan,et al.  Recent advances in trust region algorithms , 2015, Mathematical Programming.

[13]  Boying Wu,et al.  A new modified nonmonotone adaptive trust region method for unconstrained optimization , 2012, Comput. Optim. Appl..

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

[15]  Neculai Andrei.  An adaptive conjugate gradient algorithm for large-scale unconstrained optimization , 2016, J. Comput. Appl. Math..

[16]  M. Powell.  CONVERGENCE PROPERTIES OF A CLASS OF MINIMIZATION ALGORITHMS , 1975 .

[17]  Ya-Xiang Yuan,et al.  Optimization theory and methods , 2006 .

[18]  Xiwen Lu,et al.  Global convergence of BFGS and PRP methods under a modified weak Wolfe–Powell line search , 2017 .

[19]  LongHei.  A SELF—ADAPTIVE TRUST REGION ALGORITHM , 2003 .

[20]  Yong Li,et al.  A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations , 2015, Journal of Optimization Theory and Applications.

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

[22]  Jinyan Fan,et al.  An improved trust region algorithm for nonlinear equations , 2011, Comput. Optim. Appl..

[23]  Zengxin Wei,et al.  The superlinear convergence analysis of a nonmonotone BFGS algorithm on convex objective functions , 2008 .

[24]  M. Powell.  A New Algorithm for Unconstrained Optimization , 1970 .

[25]  Xiwen Lu,et al.  A conjugate gradient method with descent direction for unconstrained optimization , 2009, J. Comput. Appl. Math..

[26]  Xiwen Lu,et al.  A modified PRP conjugate gradient method , 2009, Ann. Oper. Res..

[27]  Xiaoping Lu,et al.  A quasi-Newton trust region method with a new conic model for the unconstrained optimization , 2008, Appl. Math. Comput..

[28]  Zhiguo Wang,et al.  A limited memory BFGS-type method for large-scale unconstrained optimization , 2008, Comput. Math. Appl..

[29]  Zengxin Wei,et al.  A Trust Region Algorithm with Conjugate Gradient Technique for Optimization Problems , 2011 .

[30]  Gonglin Yuan,et al.  Modified nonlinear conjugate gradient methods with sufficient descent property for large-scale optimization problems , 2009, Optim. Lett..

[31]  Qin Ni,et al.  A new regularized quasi-Newton algorithm for unconstrained optimization , 2015, Appl. Math. Comput..

[32]  Gonglin Yuan,et al.  Convergence analysis of a modified BFGS method on convex minimizations , 2010, Comput. Optim. Appl..

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

[34]  Guoyin Li,et al.  A modified Polak-Ribière-Polyak conjugate gradient algorithm for nonsmooth convex programs , 2014, J. Comput. Appl. Math..

[35]  Elizabeth Eskow,et al.  A New Modified Cholesky Factorization , 1990, SIAM J. Sci. Comput..

[36]  Masoud Ahookhosh,et al.  Computers and Mathematics with Applications a Nonmonotone Trust Region Method with Adaptive Radius for Unconstrained Optimization Problems , 2022 .

[37]  T. Steihaug.  The Conjugate Gradient Method and Trust Regions in Large Scale Optimization , 1983 .

[38]  Li Gai-di.  A Trust Region Method with Automatic Determination of the Trust Region Radius , 2006 .

[39]  Qingying Sun,et al.  A new non-monotone self-adaptive trust region method for unconstrained optimization , 2011 .