Abstract
Given a real (finite-dimensional or infinite-dimensional) Hilbert space H with a Jordan product, we consider the Lorentz cone linear complementarity problem, denoted by LCP(T, Ω, q), where T is a continuous linear operator on H, Ω ⊂ H is a Lorentz cone, and q ∈ H. We investigate some conditions for which the problem concerned has a unique solution for all q ∈ H (i.e., T has the GUS-property). Several sufficient conditions and several necessary conditions are given. In particular, we provide two sufficient and necessary conditions of T having the GUS-property. Our approach is based on properties of the Jordan product and the technique from functional analysis, which is different from the pioneer works given by Gowda and Sznajder (2007) in the case of finite-dimensional spaces.
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References
Borwein J M. Generalized linear complementarity problems treated without fixed-point theory. J Optim Theory Appl, 1984, 43: 343–356
Borwein J M, Dempster M A H. The linear order complementarity problem. Math Oper Res, 1989, 14: 534–558
Chiang Y Y, Pan S H, Chen J S. A merit function method for infinite-dimensional SOCCPs. Preprint, 2010
Cottle R W, Pang J S, Stone R E. The Linear Complementarity Problem. Boston: Academic Press, 1992
Dash A T, Nanda S. A complementarity problem in mathematical programming in Banach space. J Math Anal Appl, 1984, 98: 318–331
Doverspike R. Some perturbation results for the linear complementarity problem. Math Program, 1982, 23: 181–192
Faraut J, Korányi A. Analysis on Symmetric Cones. Oxford Mathematical Monographs. New York: Oxford University Press, 1994
Gowda M S, Seidman T I. Generalized linear complementarity problem. Math Program, 1990, 46: 329–340
Gowda M S, Sznajder R. Some global uniqueness and solvability results for linear complementarity problems over symmetric cones. SIAM J Optim, 2007, 18: 461–481
Gowda M S, Sznajder R, Tao J. Some P-properties for linear transformations on Euclidean Jordan algebras. Linear Algebra Appl, 2004, 393: 203–232
Habetler G J, Price A L. Existence theory for generalized nonlinear complementarity problems. J Optim Theory Appl, 1971, 7: 223–239
Huang Z H. Sufficient conditions on nonemptiness and boundedness of the solution set of the P 0 function nonlinear complementarity problem. Oper Res Lett, 2002, 30: 202–210
Huang Z H, Han J, Xu D C, et al. The non-interior continuation methods for solving the P 0-function nonlinear complementarity problem. Sci China Ser A, 2001, 44: 1107–1114
Isac G. Complementarity Problems. In: Lecture Notes in Mathematics, vol. 1528. Berlin: Springer, 1992
Karamardian S. Generalized complementarity problem. J Optim Theory Appl, 1971, 8: 161–168
Karamardian S. The complementarity problem. Math Program, 1972, 2: 107–129
Kreyszig E. Introductory Functional Analysis with Applications. New York: John Wiley & Sons, 1978
Miao X H, Huang Z H. The column-sufficiency and row-sufficiency of the linear transformation on Hilbert spaces. J Global Optim, 2010, 49: 109–123
Miao X H, Huang Z H, Han J. Some w-unique and w-P properties for linear transformations on Hilbert spaces. Acta Math Appl Sin Engl Ser, 2010, 26: 23–32
Wilhelm K. Jordan algebras and holomorphy. In: Functional Analysis, Holomorphy, and Approximation Theory. Lecture Notes in Mathematics, vol. 843. Berlin: Springer, 1981, 341–365
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Miao, X., Huang, Z. GUS-property for Lorentz cone linear complementarity problems on Hilbert spaces. Sci. China Math. 54, 1259–1268 (2011). https://doi.org/10.1007/s11425-011-4169-x
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DOI: https://doi.org/10.1007/s11425-011-4169-x