A nonlinear theorem of the alternative without regularity assumption (Q917854)

It is supposed that Y and Z are real topological vector spaces with Y being locally convex; \(P\subset Y\) and \(Q\subset Z\) are nonvoid convex cones with P closed and int \(Q\neq \emptyset\); \(P^*\) and \(Q^*\) are the polar cones of P and Q, respectively; \(S\subset Y\times Z\) is a nonvoid convex set. Under these assumptions the authors prove the equivalence of the following two statements: (i) the system (y,z)\(\in S\), \(y\in -P\), \(z\in int(-Q)\) has no solution; (ii) there exists \((z^*,t^*)\in Q^*\times R_+\setminus \{(0,0)\}\) such that for all \(\epsilon >0\) and for all finite subsets \({\mathcal W}\subset S\) there exists \(y^*\in P^*\) satisfying \[ t^*-\epsilon \leq <y^*,y>:<z^*,z>\text{ for all } (y,z)\in {\mathcal W}. \] Furthermore, they derive under various additional regularity assumptions several equivalent versions of statement (ii). Finally, the obtained results are used to characterize weakly efficient points.