Abstract
Motivated by the equivalence of the strict semimonotonicity property of the matrix A and the uniqueness of the solution to the linear complementarity problem LCP(A,q) for q∈R n+ , we study the strict semimonotonicity (SSM) property of linear transformations on Euclidean Jordan algebras. Specifically, we show that, under the copositive condition, the SSM property is equivalent to the uniqueness of the solution to LCP(L,q) for all q in the symmetric cone K. We give a characterization of the uniqueness of the solution to LCP(L,q) for a Z transformation on the Lorentz cone ℒ n+ . We study also a matrix-induced transformation on the Lorentz space ℒn.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)
Gowda, M.S., Sznajder, R., Tao, J.: Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Its Appl. 393, 203–232 (2004)
Gowda, M.S., Song, Y.: On semidefinite linear complementarity problems. Math. Program., Ser. A 88, 575–587 (2000)
Tao, J., Gowda, M.S.: Some P-properties for nonlinear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 985–1004 (2005)
Gowda, M.S., Tao, J.: Z-transformations on proper and symmetric cones. Math. Program., Ser. B 117, 195–221 (2009)
Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford University Press, Oxford (1994)
Schmieta, S.H., Alizadeh, F.: Extension of primal-dual interior point algorithms to symmetric cones. Math. Program. Ser. A 96, 409–438 (2003)
Facchinei, F., Pang, J.S.: Finite Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
Karamardian, S.: An existence theorem for the complementarity problem. J. Optim. Theory Appl. 19, 227–232 (1976)
Malik, M.: Some geometrical aspects of the cone linear complementarity problem. PhD Thesis, Indian Statistical Institute, New Delhi (2004)
Gowda, M.S., Sznajder, R.: Some global uniqueness and solvability results for linear complementarity problems over symmetric cones. SIAM J. Optim. 18, 461–481 (2007)
Raman, D.S.: Some contributions to semidefinite linear complementarity problem. PhD Thesis, India Statical Institute (2003)
Gowda, M.S., Parthasarathy, T.: Complementarity forms of theorems of Lyapunov and Stein, and related results. Linear Algebra Appl. 320, 131–144 (2000)
Tao, J.: Positive principal minor property of linear transformations on Euclidean Jordan algebras. J. Optim. Theory Appl. 140, 131–152 (2009)
Alizadeh, F., Goldfarb, D.: Second-order cone programming. Math. Program., Ser. B 95, 3–51 (2003)
Gowda, M.S., Sznajder, R.: Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 109–123 (2006)
Tao, J.: Some P-properties for linear transformations on the Lorentz cone. PhD Thesis, University of Maryland Baltimore County (2004)
Gowda, M.S., Song, Y.: Semidefinite relaxations of linear complementarity problems. Technical Report, Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250 (2002)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F.A. Potra.
The work is supported by the 2008 Summer Research Grant of Loyola College in Maryland. The author is grateful to Professor M.S. Gowda and to the referees for their helpful suggestions.
Rights and permissions
About this article
Cite this article
Tao, J. Strict Semimonotonicity Property of Linear Transformations on Euclidean Jordan Algebras. J Optim Theory Appl 144, 575–596 (2010). https://doi.org/10.1007/s10957-009-9611-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-009-9611-7