pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multi-plier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed p-th power Lagrangian method convexiies the perturbation function and guarantees the existence of an optimal generating Lagrangian multi-plier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimalsolution of the primal problem via the maximiza-tion of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for special cases with a single Lagrangian constraint, while adopting the p-th power Lagrangian method.