A generalization of the Gronwall-Bellman lemma and its applications

In this study we present an important theorem of the alternative involving convex functions and convex cones. From this theorem we develop saddle value optimality criteria and stationary optimality criteria for convex programs. Under suitable constraint qualification we obtain a generalized form of the Kuhn-Tucker conditions. We also use the theorem of the alternative in developing an important duality theorem. No duality gaps are encountered under the constraint qualification imposed earlier and the dual problem always possesses a solution. Moreover, it is shown that all constraint qualifications assure that the primal problem is stable in the sense used by Gale and others. The notion of stability is closely tied up with the positivity of the lagrangian multiplier of the objective function.