Frontiers of Mathematics in China >
Nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization
Received date: 21 Feb 2013
Accepted date: 29 Nov 2013
Published date: 26 Aug 2014
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We propose a nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization. Unlike traditional trust region methods, the subproblem in our method is a simple conic model, where the Hessian of the objective function is approximated by a scalar matrix. The trust region radius is adjusted with a new self-adaptive adjustment strategy which makes use of the information of the previous iteration and current iteration. The new method needs less memory and computational efforts. The global convergence and Q-superlinear convergence of the algorithm are established under the mild conditions. Numerical results on a series of standard test problems are reported to show that the new method is effective and attractive for large scale unconstrained optimization problems.
Lijuan ZHAO , Wenyu SUN , Raimundo J. B. de SAMPAIO . Nonmonotone adaptive trust region method based on simple conic model for unconstrained optimization[J]. Frontiers of Mathematics in China, 2014 , 9(5) : 1211 -1238 . DOI: 10.1007/s11464-014-0356-8
| 1 |
Andrei N. Scaled conjugate gradient algorithms for unconstrained optimization. Comput Optim Appl, 2007, 38: 401-416
|
| 2 |
Andrei N. An unconstrained optimization test functions collection. Adv Model Optim, 2008, 10: 147-161
|
| 3 |
Ariyawansa K A. Deriving collinear scaling algorithms as extensions of quasi-Newton methods and the local convergence of DFP and BFGS related collinear scaling algorithm. Math Program, 1990, 49: 23-48
|
| 4 |
Ariyawansa K A, Lau D T M. Local and Q-superlinear convergence of a class of collinear scaling algorithms that extends quasi-Newton methods with Broyden’s bounded-ϕ class of updates. Optimization, 1992, 23: 323-339
|
| 5 |
Conn A R, Gould N I M, Toint Ph L. Trust-Region Methods. Philadelphia: SIAM, 2000
|
| 6 |
Cui Z C, Wu B Y, Qu S J. Combining nonmonotone conic trust region and line search techniques for unconstrained optimization. J Comput Appl Math, 2011, 235: 2432-2441
|
| 7 |
Dai Y H. A nonmonotone conjugate gradient algorithm for unconstrained optimization. J Syst Sci Complex, 2002, 15: 139-145
|
| 8 |
Dai Y H. On the nonmonotone line search. J Optim Theory Appl, 2002, 112: 315-330
|
| 9 |
Davidon W C. Conic approximations and collinear scalings for optimizers. SIAM J Numer Anal, 1980, 17: 268-281
|
| 10 |
Deng N Y, Li Z F. Some global convergence properties of a conic variable metric algorithm for minimization with inexact line search. Optim Methods Softw, 1995, 5: 105-122
|
| 11 |
Di S, Sun W. A trust region method for conic model to solve unconstrained optimizations. Optim Methods Softw, 1996, 6: 237-263
|
| 12 |
Dolan E D, Moré J J. Benchmarking optimization software performance profiles. Math Program, 2002, 91: 201-213
|
| 13 |
Fu J H, Sun W, de Sampaio R J B. An adaptive approach of conic trust region method for unconstrained optimization problems. J Appl Math Comput, 2005, 19: 165-177
|
| 14 |
Gourgeon H, Nocedal J. A conic algorithm for optimization. SIAM J Sci Stat Comput, 1985, 6: 253-267
|
| 15 |
Grippo L, Lampariello F, Lucidi S. A nonmonotone line search technique for Newton’s method. SIAM J Numer Anal, 1986, 23: 707-716
|
| 16 |
Grippo L, Sciandrone M. Nonmonotone globalization techniques for the Barzilai-Borwein gradient method. Comput Optim Appl, 2002, 23: 143-169
|
| 17 |
Han Q, Sun W, Han J, Sampaio R J B. An adaptive trust-region method for unconstrained optimization. Optim Methods Softw, 2005, 20: 665-677
|
| 18 |
Ji Y, Li Y J, Zhang K C, Zhang X L. A new nonmonotone trust-region method of conic model for solving unconstrained optimization. J Comput Appl Math, 2010, 233: 1746-1754
|
| 19 |
Moré J J. Recent developments in algorithms and software for trust region methods. In: Bachem A, Grötschel M, Korte B, eds. Mathematical Programming: The State of the Art. Berlin: Springer, 1983, 258-287
|
| 20 |
Ni Q. Optimality conditions for trust region subproblems involving a conic model. SIAM J Optim, 2005, 15: 828-837
|
| 21 |
Nocedal J, Wright S J. Numerical Optimization. New York: Springer, 1999
|
| 22 |
Powell M J D. A new algorithm for unconstrained optimization. In: Rosen J B, Mangasarian O L, Ritter K, eds. Nonlinear Programming. New York: Academic Press, 1970, 31-65
|
| 23 |
Powell M J D, Toint Ph L. On the estimation of sparse Hessian matrices. SIAM J Numer Anal, 1979, 16: 1060-1074
|
| 24 |
Powell M J D, Yuan Y. A trust region algorithm for equality constrained optimization. Math Program, 1991, 49: 189-211
|
| 25 |
Sang Z Y, Sun Q Y. A self-adaptive trust region method with line search based on a simple subproblem model. J Comput Appl Math, 2009, 232: 514-522
|
| 26 |
Schubert L K. Modification of a quasi-Newton method for nonlinear equations with sparse Jacobian. Math Comp, 1970, 24: 27-30
|
| 27 |
Sorensen D C. The Q-superlinear convergence of a collinear scaling algorithm for unconstrained optimization. SIAM J Numer Anal, 1980, 17: 84-114
|
| 28 |
Sun Q Y, Duan L N, Cui B. A nonmonotone trust region algorithm with simple quadratic models. J Systems Sci Math Sci, 2009, 29: 470-483
|
| 29 |
Sun W. Optimization methods for non-quadratic model. Asia-Pac J Oper Res, 1996, 13: 43-63
|
| 30 |
Sun W. Nonmonotone trust region method for solving optimization problems. Appl Math Comput, 2004, 156: 159-174
|
| 31 |
Sun W, Han J, Sun J. Global convergence of nonmonotone descent methods for unconstrained optimization problems. J Comput Appl Math, 2002, 146: 89-98
|
| 32 |
Sun W, Xu D. A filter-trust-region method based on conic model for unconstrained optimization. Sci China Math, 2012, 42: 527-543 (in Chinese)
|
| 33 |
Sun W, Yuan Y. Optimization Theory and Methods: Nonlinear Programming. New York: Springer, 2006
|
| 34 |
Toint Ph L. Towards an efficient sparsity exploiting Newton method for minimization. In: Duff I S, ed. Sparse Matrices and Their Uses. New York: Academic Press, 1981, 57-88
|
| 35 |
Toint Ph L. An assessment of non-monotone line search techniques for unconstrained optimization. SIAM J Sci Comput, 1996, 17: 725-739
|
| 36 |
Xu C X, Yang X Y. Convergence of conic-quasi-Newton trust-region methods for unconstrained optimization. Math Appl (Wuhan), 1998, (2): 71-76 (in Chinese)
|
| 37 |
Zhang H C, Hager W W. A nonmonotone line search technique and its application to unconstrained optimization. SIAM J Optim, 2004, 14: 1043-1056
|
| 38 |
Zhou Q Y, Zhang C. A new nonmonotone adaptive trust region method based on simple quadratic models. J Appl Math Comput, 2012, 40: 111-123
|
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