Implied constraints and an alternate unified development of nonlinear programming theory

Abstract

This paper offers an alternate unified view of nonlinear programming theory from the perspective of implied constraints. Optimality is identically characterized for both constrained and unconstrained problems in terms of implied constraints. It is shown that there is a weaker condition than the Guignard constraint qualification for the existence of finite multipliers in the Karush-Kuhn-Tucker conditions. Surprisingly, this condition does not directly qualify the constraints but instead qualifies the objective in terms of implied constraints. More surprisingly, the existence of the finite multipliers follows directly from this objective qualification — it is not necessary for the point to be a local optimum. Methods for generating implied constraints are used to obtain a more general sufficient condition for local and global optimality. A single unified formulation of duality shows that duality is nothing more than an effort to generate the tightest implied constraint. Duality theorems hold in general for this formulation — convexity is not required — and the existence of the duality gap in prior formulations is easily explained. The algorithmic potential of this approach is highlighted by showing that the Simplex method systematically tries to imply the objective from the constraints of the problem.