Alternative theorems for general complementarity problems. (English) Zbl 0591.90088

Borwein, Jonathan M.

Infinite programming, Proc. Int. Symp., Cambridge/U.K. 1984, Lect. Notes Econ. Math. Syst. 259, 194-203 (1985).

[For the entire collection see Zbl 0569.00009.]
Let X, Y be a pair of topological vector spaces with an associated bilinear form \(<.,.>\). For a given closed convex cone \(S\subset X\), and a mapping \(T: S\to Y\), the topological complementarity problem, TCP(v), is to find a solution x to: \(<T(x)+v\), \(x>=0\), \(x\in S\), \(T(x)+v\in S^+\); for \(v\in Y\). Here \(S^+\) denotes the dual cone to S in Y. The author has proved a convex alternative theorem. He has then applied it in several different frameworks, to establish the existence of solutions to the TCP.

Reviewer: K.G.Murty

MSC:

90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C48	Programming in abstract spaces
49J27	Existence theories for problems in abstract spaces

Keywords:

monotonicity; copositivity; coercivity; P matrices; order; complementarity; bilinear form; topological complementarity problem; convex alternative theorem; existence of solutions