The algorithms of Broyden-CG for unconstrained optimization problems
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| internalnotes | [1] Nocedal, J. and S. J. Wright, Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006: Springer. [2] Byrd, R. H. and F. Nocedal, A Tool for the Analysis of Quasi-Newton Methods with Application to Unconstrained Minimization. SIAM Journal on Numerical Analysis, 1989. 26 (3): p. 727 - 739. http://dx.doi.org/10.1137/0726042 [3] Armijo, L., Minimization of functions having Lipschitz continuous partial derivatives. Pacific Journal of Mathematics, 1966. 16 (1): p. 1 - 3. http://dx.doi.org/10.2140/pjm.1966.16.1 [4] Han, L. and M. Neumann, Combining quasi-Newton and Cauchy directions. International Journal of Applied Mathematics, 2003. 12 (2): p. 167 - 191. [5] Xu, D.-c., Global Convergence of the Broyden's Class of Quasi-Newton Methods with Nonmonotone Linesearch. Acta Mathematicae Applicatae Sinica, 2003. 19 (1): p. 19 - 24. http://dx.doi.org/10.1007/s10255-003-0076-4 [6] Byrd, R. H., J. Nocedal, and Y.-X. Yuan, Global Convergence of a Class of Quasi-Newton Methods on Convex Problems. SIAM Journal on Numerical Analysis, 1987. 24 (5): p. 1171 - 1191. http://dx.doi.org/10.1137/0724077 [7] Chong, E. K. P. and S. H. Zak, An Introduction to Optimization. Second ed. 2001: A Wiley-Interscience Publication. [8] Ludwig, A., The Gauss–Seidel–quasi-Newton method: A hybrid algorithm for solving dynamic economic models. Journal of Economic Dynamics and Control, 2007. 31 (5): p. 1610 - 1632. http://dx.doi.org/10.1016/j.jedc.2006.05.007 [9] Luo, Y.-Z., G.-J. Tang, and L.-N. Zhou, Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method. Applied Soft Computing, 2008.8 (2): p.1068 - 1073. http://dx.doi.org/10.1016/j.asoc.2007.05.013 [10] Ibrahim, M. A. H., M. Mamat, and W. J. Leong, The Hybrid BFGS-CG Method in Solving Unconstrained Optimization Problems. Abstract and Applied Analysis, 2014. 2014: p. 6. http://dx.doi.org/10.1155/2014/507102 [11] Ibrahim, M. A. H., et al., Alternative Algorithms Of Broyden Familyami: For Unconstrained Optimization. AIP Conference Proceedings, 2010. 1309 (1): p. 670-680. http://dx.doi.org/10.1063/1.3525191 [12] Ibrahim, M. A. H. B., et al., The CG-BFGS method for unconstrained optimization problems. AIP Conference Proceedings, 2014. 1605: p. 167 - 172. http://dx.doi.org/10.1063/1.4887583 [13] Mamat, M., et al., Hybrid Broyden Method for Unconstrained Optimization. International Journal of Numerical Methods and Applications, 2009. 1 (2): p. 121-130. [14] Powell, M. J. D., Restart procedures for the conjugate gradient method. Mathematical Programming, 1977. 12 (1): p. 241 - 254. http://dx.doi.org/10.1007/bf01593790 [15] Andrei, N., An Unconstrained Optimization Test Functions Collection. Advanced Modelling and Optimization, 2008. 10 (1): p. 147 - 161. [16] Zbigniew Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs. 1996: Springer Verlag. http://dx.doi.org/10.1007/978-3-662-03315-9 [17] More, J. J., B. S. Garbow, and K. E. Hillstrom, Testing Unconstrained Optimization Software. ACM Trans. Math. Softw., 1981. 7 (1): p. 17 - 41. http://dx.doi.org/10.1145/355934.355936 [18] Dolan, E. D. and J. J. Moré, Benchmarking optimization software with performance profiles. Mathematical Programming, 2002. 91 (2): p. 201 - 213. http://dx.doi.org/10.1007/s101070100263 |
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| spelling | 11399 https://intelek.unisza.edu.my/intelek/pages/view.php?ref=11399 https://intelek.unisza.edu.my/intelek/pages/search.php?search=!collection407072 Restricted Document Article Journal UniSZA Unisza unisza image/jpeg inches 796 96 96 1426 51 51 2015-02-18 10:59:49 1426x796 5632-01-FH02-FIK-15-02561.jpg UniSZA Private Access The algorithms of Broyden-CG for unconstrained optimization problems International Journal of Mathematical Analysis The conjugate gradient method plays an important role in solving large-scaled problems and the quasi-Newton method is known as the most efficient method in solving unconstrained optimization problems. Therefore, in this paper, the new hybrid method between the conjugate gradient method and the quasi-newton method for solving optimization problem is suggested. The Broyden family formula is used as an approximation of Hessian in the hybrid method and the quasi-Newton method. Our numerical analysis provides strong evidence that our Broyden-CG method is more efficient than the ordinary Broyden method. Furthermore, we also prove that new algorithm is globally convergent and gratify the sufficient descent condition. 8 45 HIKARI Ltd. HIKARI Ltd. 2591-2600 [1] Nocedal, J. and S. J. Wright, Numerical Optimization. Springer Series in Operations Research and Financial Engineering. 2006: Springer. [2] Byrd, R. H. and F. Nocedal, A Tool for the Analysis of Quasi-Newton Methods with Application to Unconstrained Minimization. SIAM Journal on Numerical Analysis, 1989. 26 (3): p. 727 - 739. http://dx.doi.org/10.1137/0726042 [3] Armijo, L., Minimization of functions having Lipschitz continuous partial derivatives. Pacific Journal of Mathematics, 1966. 16 (1): p. 1 - 3. http://dx.doi.org/10.2140/pjm.1966.16.1 [4] Han, L. and M. Neumann, Combining quasi-Newton and Cauchy directions. International Journal of Applied Mathematics, 2003. 12 (2): p. 167 - 191. [5] Xu, D.-c., Global Convergence of the Broyden's Class of Quasi-Newton Methods with Nonmonotone Linesearch. Acta Mathematicae Applicatae Sinica, 2003. 19 (1): p. 19 - 24. http://dx.doi.org/10.1007/s10255-003-0076-4 [6] Byrd, R. H., J. Nocedal, and Y.-X. Yuan, Global Convergence of a Class of Quasi-Newton Methods on Convex Problems. SIAM Journal on Numerical Analysis, 1987. 24 (5): p. 1171 - 1191. http://dx.doi.org/10.1137/0724077 [7] Chong, E. K. P. and S. H. Zak, An Introduction to Optimization. Second ed. 2001: A Wiley-Interscience Publication. [8] Ludwig, A., The Gauss–Seidel–quasi-Newton method: A hybrid algorithm for solving dynamic economic models. Journal of Economic Dynamics and Control, 2007. 31 (5): p. 1610 - 1632. http://dx.doi.org/10.1016/j.jedc.2006.05.007 [9] Luo, Y.-Z., G.-J. Tang, and L.-N. Zhou, Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method. Applied Soft Computing, 2008.8 (2): p.1068 - 1073. http://dx.doi.org/10.1016/j.asoc.2007.05.013 [10] Ibrahim, M. A. H., M. Mamat, and W. J. Leong, The Hybrid BFGS-CG Method in Solving Unconstrained Optimization Problems. Abstract and Applied Analysis, 2014. 2014: p. 6. http://dx.doi.org/10.1155/2014/507102 [11] Ibrahim, M. A. H., et al., Alternative Algorithms Of Broyden Familyami: For Unconstrained Optimization. AIP Conference Proceedings, 2010. 1309 (1): p. 670-680. http://dx.doi.org/10.1063/1.3525191 [12] Ibrahim, M. A. H. B., et al., The CG-BFGS method for unconstrained optimization problems. AIP Conference Proceedings, 2014. 1605: p. 167 - 172. http://dx.doi.org/10.1063/1.4887583 [13] Mamat, M., et al., Hybrid Broyden Method for Unconstrained Optimization. International Journal of Numerical Methods and Applications, 2009. 1 (2): p. 121-130. [14] Powell, M. J. D., Restart procedures for the conjugate gradient method. Mathematical Programming, 1977. 12 (1): p. 241 - 254. http://dx.doi.org/10.1007/bf01593790 [15] Andrei, N., An Unconstrained Optimization Test Functions Collection. Advanced Modelling and Optimization, 2008. 10 (1): p. 147 - 161. [16] Zbigniew Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs. 1996: Springer Verlag. http://dx.doi.org/10.1007/978-3-662-03315-9 [17] More, J. J., B. S. Garbow, and K. E. Hillstrom, Testing Unconstrained Optimization Software. ACM Trans. Math. Softw., 1981. 7 (1): p. 17 - 41. http://dx.doi.org/10.1145/355934.355936 [18] Dolan, E. D. and J. J. Moré, Benchmarking optimization software with performance profiles. Mathematical Programming, 2002. 91 (2): p. 201 - 213. http://dx.doi.org/10.1007/s101070100263 |
| spellingShingle | The algorithms of Broyden-CG for unconstrained optimization problems |
| summary | The conjugate gradient method plays an important role in solving large-scaled problems and the quasi-Newton method is known as the most efficient method in solving unconstrained optimization problems. Therefore, in this paper, the new hybrid method between the conjugate gradient method and the quasi-newton method for solving optimization problem is suggested. The Broyden family formula is used as an approximation of Hessian in the hybrid method and the quasi-Newton method. Our numerical analysis provides strong evidence that our Broyden-CG method is more efficient than the ordinary Broyden method. Furthermore, we also prove that new algorithm is globally convergent and gratify the sufficient descent condition. |
| title | The algorithms of Broyden-CG for unconstrained optimization problems |
| title_full | The algorithms of Broyden-CG for unconstrained optimization problems |
| title_fullStr | The algorithms of Broyden-CG for unconstrained optimization problems |
| title_full_unstemmed | The algorithms of Broyden-CG for unconstrained optimization problems |
| title_short | The algorithms of Broyden-CG for unconstrained optimization problems |
| title_sort | algorithms of broyden-cg for unconstrained optimization problems |