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 & 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. doi: 10.1007/s10589-009-9297-7. Google Scholar

[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. doi: 10.1016/j.nonrwa.2010.09.021. Google Scholar

[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. doi: 10.1090/S0025-5718-1974-0343581-1. Google Scholar

[4]

A. Fischer, A special Newton-type optimization method,, Optim., 24 (1992), 269. doi: 10.1080/02331939208843795. Google Scholar

[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. doi: 10.1007/s11075-008-9194-7. Google Scholar

[6]

J. J. Júdice, H. D. Sherali and I. M. Ribeiro, The eigenvalue complementarity problem,, Comput. Optim. Appl., 37 (2007), 139. doi: 10.1007/s10589-007-9017-0. Google Scholar

[7]

D. H. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems,, Comput. Optim. Appl., 17 (2000), 203. doi: 10.1023/A:1026502415830. Google Scholar

[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. doi: 10.1023/A:1012996232707. Google Scholar

[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. doi: 10.1080/10556780008805782. Google Scholar

[10]

J. J. Moré and A. Trangenstein, On the global convergence of Broyden's method,, Math. Comput., 30 (1976), 523. Google Scholar

[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. doi: 10.1016/j.cma.2003.09.013. Google Scholar

[12]

A. Pinto da Costa and A. Seeger, Numerical resolution of cone constrained eigenvalue problems,, Comput. Appl. Math., 28 (2009), 37. doi: 10.1590/S0101-82052009000100003. Google Scholar

[13]

A. Pinto da Costa and A. Seeger, Cone-constrained eigenvalue problems: theory and algorithms,, Comput. Optim. Appl., 45 (2010), 25. doi: 10.1007/s10589-008-9167-8. Google Scholar

[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. Google Scholar

[15]

A. Seeger and M. Torki, On eigenvalues induced by a cone constraint,, Linear Algebra Appl., 372 (2003), 181. doi: 10.1016/S0024-3795(03)00553-6. Google Scholar

show all references

References:
[1]

S. Adly and A. Seeger, A nonsmooth algorithm for cone constrained eigenvalue problems,, Comput. Optim. Appl., 49 (2011), 299. doi: 10.1007/s10589-009-9297-7. Google Scholar

[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. doi: 10.1016/j.nonrwa.2010.09.021. Google Scholar

[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. doi: 10.1090/S0025-5718-1974-0343581-1. Google Scholar

[4]

A. Fischer, A special Newton-type optimization method,, Optim., 24 (1992), 269. doi: 10.1080/02331939208843795. Google Scholar

[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. doi: 10.1007/s11075-008-9194-7. Google Scholar

[6]

J. J. Júdice, H. D. Sherali and I. M. Ribeiro, The eigenvalue complementarity problem,, Comput. Optim. Appl., 37 (2007), 139. doi: 10.1007/s10589-007-9017-0. Google Scholar

[7]

D. H. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems,, Comput. Optim. Appl., 17 (2000), 203. doi: 10.1023/A:1026502415830. Google Scholar

[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. doi: 10.1023/A:1012996232707. Google Scholar

[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. doi: 10.1080/10556780008805782. Google Scholar

[10]

J. J. Moré and A. Trangenstein, On the global convergence of Broyden's method,, Math. Comput., 30 (1976), 523. Google Scholar

[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. doi: 10.1016/j.cma.2003.09.013. Google Scholar

[12]

A. Pinto da Costa and A. Seeger, Numerical resolution of cone constrained eigenvalue problems,, Comput. Appl. Math., 28 (2009), 37. doi: 10.1590/S0101-82052009000100003. Google Scholar

[13]

A. Pinto da Costa and A. Seeger, Cone-constrained eigenvalue problems: theory and algorithms,, Comput. Optim. Appl., 45 (2010), 25. doi: 10.1007/s10589-008-9167-8. Google Scholar

[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. Google Scholar

[15]

A. Seeger and M. Torki, On eigenvalues induced by a cone constraint,, Linear Algebra Appl., 372 (2003), 181. doi: 10.1016/S0024-3795(03)00553-6. Google Scholar

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