Limiting Lagrangians: a primal approach

The authors consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. They show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vector d with the following properties: First, whenever epsilon d is added to the objective function, where epsilon is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as epsilon tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program.