Theorems of the alternative and their applications in multiobjective optimization (Q1064972)

Let \(A: R^ n\to R^ n\), \(B: R^ n\to R^ m\) be linear maps. Let M be a convex cone in \(R^ k\), N be a convex cone in \(R^ m\). By using a separation theorem the author provides three theorems of the alternative. The most important theorem asserts that under the closedness of \(A(B^{- 1}(N))\), either there is a vector \(z\in R^ n\) with Az'\(\in ri M\) and Bz'\(\in N\), or there is a nontrivial vector \(\lambda\) from the nonnegative polar cone \(M^*\) of M such that -A'\(\lambda\) '\(\in cl B'(N^*)\), but never both. This result is then applied to derive several optimality conditions (including the well-known Pareto optimality condition of Smale) for multiobjective programming problems. Some duality theorems are developed for the multiobjective (nonconvex) programs which yielding certain differentiability assumptions.