Asymptotic Optimality Conditions

Abstract

Optimality of a feasible point in convex programming has been characterized in Section 3.4 using the minimal index set of active constraints, i.e., using the constraints that are equal to zero on the entire feasible set. Now a different strategy will be used. Optimality will be characterized after identifying, at a point tested for optimality, the constraints that have a “locally flat surface”. If all active constraints have locally flat surfaces, then optimality is characterized by the Karush-Kuhn-Tucker conditions. If at least one active constraint does not have this property, then optimality conditions generally assume an asymptotic form (i.e., they require a closure condition). We begin the chapter by introducing differentiable convex functions with locally flat surfaces. The class will be extended to non-smooth functions in Chapter 5.