Dorn’s duality for quadratic programs revisited: The nonconvex case

The duality theory for quadratic programming problems are developed by Dorn is based principally on the assumption of convexity. The paper presents a derivation for this ‘dual’ program using the Karush-Kuhn-Tucker conditions, and examine the relationships between the pair of problems for a general nonconvex situation. Under convexity assumptions, the results of Dorn are readily recovered. however, for the nonconvex case, this pair of problems exhibits certain anomalies which preclude their interpretation as a pair of dual programs in the usual sense. These anomalies are clarified by showing that a particular Lagrangian dual formulation for the nonconvex case recovers Dorn’s dual program with an additional term in the objective function, which happens to be zero under convexity assumptions. A sufficient condition for the absence of a duality gap with respect to this revised dual problem is also presented for the nonconvex situation.