Continuity of the solution set of homogeneous equilibrium problems and linear complementarity problems
Denote by S(M, q) the solution set of the linear complementarity problem x {>=} 0, Mx + q {>=} 0, [x, Mx + q] = 0, where M {element_of} R{sup n{times}n} and q {element_of} R{sup n}. M is called an R{sub o}-matrix iff S(M, 0) = {l_brace}0{r_brace}. Jansen and Tijs have proved that if M is an R{sub 0}-matrix, then the map S is upper semicontinuous at (m, q) for every q {element_of} R{sup n}. We prove that this property is characteristic for R{sub 0}-matrices. The set of all the pairs (M, q) at which S is upper semicontinuous, is investigated in detail. Part of our results extends to homogeneous equilibrium problems of the type x {element_of} K, f(x, y) + [q, y - x] {>=} 0 {forall}y {element_of} K. Here K {improper_subset} R{sup n} is a closed convex cone and f : K {times} K {yields} R is such that f({lambda}x, {lambda}y) = {lambda}{sup +1} f(x, y) {forall}y {element_of} K, {forall}{lambda} {>=} 0, where > 0 is a fixed constant. Setting f(x, y) = [M(x), y - x], where M({center_dot}) : K {yields} R{sup n} is positively homogeneous of degree, we obtain as special case results for the homogeneous complementarity problem x {element_of} K, M(x) + q {element_of} K*, [x, M(x) + q] = 0, where K* = {l_brace}{xi} {element_of} R{sup n} : [x, {xi}] {>=} 0 {forall}x {element_of} K{r_brace} is the polar cone of K.
- OSTI ID:
- 36350
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0690
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
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