The symmetric eigenvalue complementarity problem

Queiroz, Marcelo; Júdice, Joaquim; Humes, Carlos, Jr.

doi:10.1090/S0025-5718-03-01614-4

The symmetric eigenvalue complementarity problem
HTML articles powered by AMS MathViewer

by Marcelo Queiroz, Joaquim Júdice and Carlos Humes Jr. PDF

Math. Comp. 73 (2004), 1849-1863 Request permission

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.

References

Giles Auchmuty, Unconstrained variational principles for eigenvalues of real symmetric matrices, SIAM J. Math. Anal. 20 (1989), no. 5, 1186–1207. MR 1009353, DOI 10.1137/0520078
Giles Auchmuty, Globally and rapidly convergent algorithms for symmetric eigenproblems, SIAM J. Matrix Anal. Appl. 12 (1991), no. 4, 690–706. MR 1121702, DOI 10.1137/0612053
Mokhtar S. Bazaraa and C. M. Shetty, Nonlinear programming, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Theory and algorithms. MR 533477
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
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, DOI 10.1007/978-1-4757-3279-5_{4}
S. Dirkse and M. Ferris, The PATH solver: a nonmonotone stabilization scheme for mixed complementarity problems, Optimization and Software, 5:123-156, 1995.
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, DOI 10.1007/BF01582255
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
M. Mongeau and M. Torki, Computing eigenelements of real symmetric matrices via optimization, Technical Report MIP 99-54, Université Paul Sabatier, Toulouse, 1999.
B. A. Murtagh and M. A. Saunders, MINOS 5.1 user’s guide, Report SOL 83-20R, Department of Operations Research, Stanford University, 1987.
K. G. Murty, Linear complementarity, linear and nonlinear programming, Sigma Series in Applied Mathematics, vol. 3, Heldermann Verlag, Berlin, 1988. MR 949214
James M. Ortega, Matrix theory, The University Series in Mathematics, Plenum Press, New York, 1987. A second course. MR 878977, DOI 10.1007/978-1-4899-0471-3
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, DOI 10.1137/1.9781611971163
Alberto Seeger, Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions, Linear Algebra Appl. 292 (1999), no. 1-3, 1–14. MR 1689301, DOI 10.1016/S0024-3795(99)00004-X