The symmetric eigenvalue complementarity problem

Authors:
Marcelo Queiroz, Joaquim Júdice and Carlos Humes, Jr.

Journal:
Math. Comp. **73** (2004), 1849-1863

MSC (2000):
Primary 90C33, 47A75; Secondary 90C30, 82B05

Published electronically:
August 20, 2003

MathSciNet review:
2059739

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper the Eigenvalue Complementarity Problem (EiCP) with real symmetric matrices is addressed. It is shown that the symmetric (EiCP) is equivalent to finding an equilibrium solution of a differentiable optimization problem in a compact set. A necessary and sufficient condition for solvability is obtained which, when verified, gives a convenient starting point for any gradient-ascent local optimization method to converge to a solution of the (EiCP). It is further shown that similar results apply to the Symmetric Generalized Eigenvalue Complementarity Problem (GEiCP). Computational tests show that these reformulations improve the speed and robustness of the solution methods.

**1.**Giles Auchmuty,*Unconstrained variational principles for eigenvalues of real symmetric matrices*, SIAM J. Math. Anal.**20**(1989), no. 5, 1186–1207. MR**1009353**, 10.1137/0520078**2.**Giles Auchmuty,*Globally and rapidly convergent algorithms for symmetric eigenproblems*, SIAM J. Matrix Anal. Appl.**12**(1991), no. 4, 690–706. MR**1121702**, 10.1137/0612053**3.**Mokhtar S. Bazaraa and C. M. Shetty,*Nonlinear programming*, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Theory and algorithms. MR**533477****4.**Françoise Chatelin,*Eigenvalues of matrices*, John Wiley & Sons, Ltd., Chichester, 1993. With exercises by Mario Ahués and the author; Translated from the French and with additional material by Walter Ledermann. MR**1232655****5.**A. Pinto da Costa, I. N. Figueiredo, J. J. Júdice, and J. A. C. Martins,*A complementarity eigenproblem in the stability analysis of finite dimensional elastic systems with frictional contact*, Complementarity: applications, algorithms and extensions (Madison, WI, 1999), Appl. Optim., vol. 50, Kluwer Acad. Publ., Dordrecht, 2001, pp. 67–83. MR**1818617**, 10.1007/978-1-4757-3279-5_4**6.**S. Dirkse and M. Ferris, The PATH solver: a nonmonotone stabilization scheme for mixed complementarity problems,*Optimization and Software*, 5:123-156, 1995.**7.**Patrick T. Harker and Jong-Shi Pang,*Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications*, Math. Programming**48**(1990), no. 2, (Ser. B), 161–220. MR**1073707**, 10.1007/BF01582255**8.**Joaquím J. Júdice,*Algorithms for linear complementarity problems*, Algorithms for continuous optimization (Il Ciocco, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 434, Kluwer Acad. Publ., Dordrecht, 1994, pp. 435–474. MR**1314218****9.**M. Mongeau and M. Torki, Computing eigenelements of real symmetric matrices via optimization,*Technical Report MIP 99-54*, Université Paul Sabatier, Toulouse, 1999.**10.**B. A. Murtagh and M. A. Saunders, MINOS 5.1 user's guide,*Report SOL 83-20R*, Department of Operations Research, Stanford University, 1987.**11.**K. G. Murty,*Linear complementarity, linear and nonlinear programming*, Sigma Series in Applied Mathematics, vol. 3, Heldermann Verlag, Berlin, 1988. MR**949214****12.**James M. Ortega,*Matrix theory*, The University Series in Mathematics, Plenum Press, New York, 1987. A second course. MR**878977****13.**Beresford N. Parlett,*The symmetric eigenvalue problem*, Classics in Applied Mathematics, vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998. Corrected reprint of the 1980 original. MR**1490034****14.**Alberto Seeger,*Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions*, Linear Algebra Appl.**292**(1999), no. 1-3, 1–14. MR**1689301**, 10.1016/S0024-3795(99)00004-X

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
90C33,
47A75,
90C30,
82B05

Retrieve articles in all journals with MSC (2000): 90C33, 47A75, 90C30, 82B05

Additional Information

**Marcelo Queiroz**

Affiliation:
Computer Science Department, University of São Paulo, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil

Email:
mqz@ime.usp.br

**Joaquim Júdice**

Affiliation:
Mathematics Department, University of Coimbra, 3000 Coimbra, Portugal

Email:
Joaquim.Judice@co.it.pt

**Carlos Humes, Jr.**

Affiliation:
Computer Science Department, University of São Paulo, Rua do Matão 1010, 05508-090 São Paulo, SP, Brazil

Email:
chumes@usp.br

DOI:
http://dx.doi.org/10.1090/S0025-5718-03-01614-4

Received by editor(s):
March 26, 2002

Received by editor(s) in revised form:
January 23, 2003

Published electronically:
August 20, 2003

Additional Notes:
The first author was supported by FAPESP Grant Nos. 97/06227-2 and 02/01351-7.

The second author was supported by FCT project POCTI/35059/MAT/2000.

Article copyright:
© Copyright 2003
American Mathematical Society