Implied constraints and a unified theory of duality in linear and nonlinear programming

Abstract

A unified theory of duality in linear and nonlinear programming can be easily derived from the perspective of implied constraints, that is, constraints implied by the given constraints of the problem. The key idea is to construct special types of implied constraints which will bound the objective function. This simple idea motivates a complete, concise and clear development of a unified theory of duality which seeks the tightest bound on the primal objective function. The derivation is intuitive and geometric and leads to a more general formulation of duality in nonlinear programming than the formulations known so far. Duality theorems are shown to hold for this formulation in general. To solve the dual, the set of implied constraints on the primal objective function needs to be generated. A duality gap exists if this set is only partially generated in a dual formulation. This is implicity the case in formulations such as the Lagrangian and surrogate duality. In linear programming, this set can be generated linearly resulting in the constraints of the dual problem. This formulation is also equivalent to a special conjugate function. The conjugate function approach to duality is formulated more for the perturbations in the parameters of the problem and has not developed this fact.