Enhanced Fritz John Conditions for Convex Programming

We consider convex constrained optimization problems, and we enhance the classical Fritz John optimality conditions to assert the existence of multipliers with special sensitivity properties. In particular, we prove the existence of Fritz John multipliers that are informative in the sense that they identify constraints whose relaxation, at rates proportional to the multipliers, strictly improves the primal optimal value. Moreover, we show that if the set of geometric multipliers is nonempty, then the minimum-norm vector of this set is informative and defines the optimal rate of cost improvement per unit constraint violation. Our assumptions are very general and, in particular, allow for the presence of a duality gap and the nonexistence of optimal solutions. In particular, for the case where there is a duality gap, we establish enhanced Fritz John conditions involving the dual optimal value and dual optimal solutions.