Abstract
Equilibria in mechanics or in transportation models are not always expressed through a system of equations, but sometimes they are characterized by means of complementarity conditions involving a convex cone. This work deals with the analysis of cone-constrained eigenvalue problems. We discuss some theoretical issues like, for instance, the estimation of the maximal number of eigenvalues in a cone-constrained problem. Special attention is paid to the Paretian case. As a short addition to the theoretical part, we introduce and study two algorithms for solving numerically such type of eigenvalue problems.
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References
Chu, K.W.E.: On multiple eigenvalues of matrices depending on several parameters. SIAM J. Numer. Anal. 27, 1368–1385 (1990)
Edelman, A., Kostlan, E.: How many zeros of a random polynomial are real? Bull. Am. Math. Soc. 32, 1–37 (1995)
Edelman, A., Kostlan, E., Shub, M.: How many eigenvalues of a random matrix are real? J. Am. Math. Soc. 7, 247–267 (1994)
Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)
Fan, M.K.H., Tsing, N.K., Verriest, E.I.: On analyticity of functions involving eigenvalues. Linear Algebra Appl. 207, 159–180 (1994)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge Univ. Press, Cambridge (1985)
Iusem, A., Seeger, A.: On convex cones with infinitely many critical angles. Optimization 56, 115–128 (2007)
Júdice, J.J., Sherali, H.D., Ribeiro, I.M.: The eigenvalue complementarity problem. Comput. Optim. Appl. 37, 139–156 (2007)
Krasnoselskii, M.A.: Two remarks on the method of successive approximations. Usp. Mat. Nauk 10, 123–127 (1955) (in Russian)
Lavilledieu, P., Seeger, A.: Existence de valeurs propres pour les systèmes multivoques: résultats anciens et nouveaux. Ann. Sci. Math. Que. 25, 47–70 (2000)
Minc, H.: Nonnegative Matrices. Wiley-Interscience, New York (1988)
Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cones mutuellement polaires. C. R. Acad. Sci. Paris 255, 238–240 (1962)
Pinto da Costa, A., Figueiredo, I.N., Júdice, J.A., Martins, J.A.C.: A complementarity eigenproblem in the stability of finite dimensional elastic systems with frictional contact. In: Ferris, M., Mangasarian, O., Pang, J.S. (eds.) Complementarity: Applications, Algorithms and Extensions. Applied Optimization Series, vol. 50, pp. 67–83. Kluwer Academic, Dordrecht (1999)
Pinto da Costa, A., Martins, J.A.C., Figueiredo, I.N., Júdice, J.J.: The directional instability problem in systems with frictional contacts. Comput. Methods Appl. Mech. Eng. 193, 357–384 (2004)
Queiroz, M., Júdice, J., Humes, C.: The symmetric eigenvalue complementarity problem. Math. Comput. 73, 1849–1863 (2004)
Quittner, P.: Spectral analysis of variational inequalities. Comment. Math. Univ. Carol. 27, 605–629 (1986)
Seeger, A.: Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions. Linear Algebra Appl. 292, 1–14 (1999)
Seeger, A., Torki, M.: On eigenvalues induced by a cone constraint. Linear Algebra Appl. 372, 181–206 (2003)
Singh, S., Watson, B., Srivastava, P.: Fixed Point Theory and Best Approximation: The KKM-Map Principle. Kluwer Academic, Dordrecht (1997)
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Pinto da Costa, A., Seeger, A. Cone-constrained eigenvalue problems: theory and algorithms. Comput Optim Appl 45, 25–57 (2010). https://doi.org/10.1007/s10589-008-9167-8
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DOI: https://doi.org/10.1007/s10589-008-9167-8