Generalized separation theorems and the Farkas' lemma

The main purpose of this paper is to extend the conventional separation theorems concerning the convex subset of R-m to generalized separation theorems concerning the convex subset of R-m x S-n. This is accomplished by the introduction of the generalized inner product in R-m X S-n. Then we derive the famous Farkas' lemma in nonlinear semidefinite programming, which may be very important for the analysis of the dual theory and optimality conditions of nonlinear semidefinite programming. (c) 2005 Elsevier Ltd. All rights reserved.