# American Institute of Mathematical Sciences

2011, 1(3): 529-537. doi: 10.3934/naco.2011.1.529

## A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem

 1 School of Management Science, Qufu Normal University, Rizhao, Shandong, China, China 2 School of Management Science, Qufu Normal University, Rizhao Shandong, 276800

Received  May 2011 Revised  August 2011 Published  September 2011

For the polyhedral cone constrained eigenvalue problem over a polyhedral cone, based on its nonsmooth transformed version and a smoothing technique, we propose a modified smoothing Broyden-like method and establish its convergence under suitable conditions. The given computational experiments show the efficiency of the proposed method.
Citation: Yafeng Li, Guo Sun, Yiju Wang. A smoothing Broyden-like method for polyhedral cone constrained eigenvalue problem. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 529-537. doi: 10.3934/naco.2011.1.529
##### References:
 [1] S. Adly and A. Seeger, A nonsmooth algorithm for cone constrained eigenvalue problems, Comput. Optim. Appl., 49 (2011), 299-318. doi: 10.1007/s10589-009-9297-7. [2] B. Chen and C. Ma, Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem, Nonlinear Analysis: Real World Applications, 12 (2011), 1250-1263. doi: 10.1016/j.nonrwa.2010.09.021. [3] J. E. Dennis Jr. and J. J. Moré, A characterization of superlinear convergence and its applications to quasi-Newton methods, Math. Comput., 28 (1974), 549-560. doi: 10.1090/S0025-5718-1974-0343581-1. [4] A. Fischer, A special Newton-type optimization method, Optim., 24 (1992), 269-284. doi: 10.1080/02331939208843795. [5] J. J. Júdice, M. Raydan, S. S. Rosa and S. A. Santos, On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm, Numer. Algorithms, 47 (2008), 391-407. doi: 10.1007/s11075-008-9194-7. [6] J. J. Júdice, H. D. Sherali and I. M. Ribeiro, The eigenvalue complementarity problem, Comput. Optim. Appl., 37 (2007), 139-156. doi: 10.1007/s10589-007-9017-0. [7] D. H. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Comput. Optim. Appl., 17 (2000), 203-230. doi: 10.1023/A:1026502415830. [8] D. H. Li and M. Fukushima, Globally convergent Broyden-like methods for semismooth equations and applications to VIP, NCP and MCP, Annals of Oper. Res., 103 (2001), 71-79. doi: 10.1023/A:1012996232707. [9] D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optim. Method & Software, 13 (2000), 181-201. doi: 10.1080/10556780008805782. [10] J. J. Moré and A. Trangenstein, On the global convergence of Broyden's method, Math. Comput., 30 (1976), 523-540. [11] A. Pinto da Costa, J. A. C. Martins, I. N. Figueiredo and J. J. J\'udice, The directional instability problem in systems with frictional contacts, Comput. Methods Appl. Mech. Eng., 193 (2004), 357-384. doi: 10.1016/j.cma.2003.09.013. [12] A. Pinto da Costa and A. Seeger, Numerical resolution of cone constrained eigenvalue problems, Comput. Appl. Math., 28 (2009), 37-61. doi: 10.1590/S0101-82052009000100003. [13] A. Pinto da Costa and A. Seeger, Cone-constrained eigenvalue problems: theory and algorithms, Comput. Optim. Appl., 45 (2010), 25-57. doi: 10.1007/s10589-008-9167-8. [14] L. Q. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35. [15] A. Seeger and M. Torki, On eigenvalues induced by a cone constraint, Linear Algebra Appl., 372 (2003), 181-206. doi: 10.1016/S0024-3795(03)00553-6.

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##### References:
 [1] S. Adly and A. Seeger, A nonsmooth algorithm for cone constrained eigenvalue problems, Comput. Optim. Appl., 49 (2011), 299-318. doi: 10.1007/s10589-009-9297-7. [2] B. Chen and C. Ma, Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem, Nonlinear Analysis: Real World Applications, 12 (2011), 1250-1263. doi: 10.1016/j.nonrwa.2010.09.021. [3] J. E. Dennis Jr. and J. J. Moré, A characterization of superlinear convergence and its applications to quasi-Newton methods, Math. Comput., 28 (1974), 549-560. doi: 10.1090/S0025-5718-1974-0343581-1. [4] A. Fischer, A special Newton-type optimization method, Optim., 24 (1992), 269-284. doi: 10.1080/02331939208843795. [5] J. J. Júdice, M. Raydan, S. S. Rosa and S. A. Santos, On the solution of the symmetric eigenvalue complementarity problem by the spectral projected gradient algorithm, Numer. Algorithms, 47 (2008), 391-407. doi: 10.1007/s11075-008-9194-7. [6] J. J. Júdice, H. D. Sherali and I. M. Ribeiro, The eigenvalue complementarity problem, Comput. Optim. Appl., 37 (2007), 139-156. doi: 10.1007/s10589-007-9017-0. [7] D. H. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Comput. Optim. Appl., 17 (2000), 203-230. doi: 10.1023/A:1026502415830. [8] D. H. Li and M. Fukushima, Globally convergent Broyden-like methods for semismooth equations and applications to VIP, NCP and MCP, Annals of Oper. Res., 103 (2001), 71-79. doi: 10.1023/A:1012996232707. [9] D. H. Li and M. Fukushima, A derivative-free line search and global convergence of Broyden-like method for nonlinear equations, Optim. Method & Software, 13 (2000), 181-201. doi: 10.1080/10556780008805782. [10] J. J. Moré and A. Trangenstein, On the global convergence of Broyden's method, Math. Comput., 30 (1976), 523-540. [11] A. Pinto da Costa, J. A. C. Martins, I. N. Figueiredo and J. J. J\'udice, The directional instability problem in systems with frictional contacts, Comput. Methods Appl. Mech. Eng., 193 (2004), 357-384. doi: 10.1016/j.cma.2003.09.013. [12] A. Pinto da Costa and A. Seeger, Numerical resolution of cone constrained eigenvalue problems, Comput. Appl. Math., 28 (2009), 37-61. doi: 10.1590/S0101-82052009000100003. [13] A. Pinto da Costa and A. Seeger, Cone-constrained eigenvalue problems: theory and algorithms, Comput. Optim. Appl., 45 (2010), 25-57. doi: 10.1007/s10589-008-9167-8. [14] L. Q. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35. [15] A. Seeger and M. Torki, On eigenvalues induced by a cone constraint, Linear Algebra Appl., 372 (2003), 181-206. doi: 10.1016/S0024-3795(03)00553-6.
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