Copositivity for second-order optimality conditions in general smooth optimization problems

Abstract

Second-order local optimality conditions involving copositivity of the Hessian of the Lagrangian on the reduced linearization cone have the advantage that there is only a small gap between sufficient (the Hessian is strictly copositive) and necessary (the Hessian is copositive) conditions. In this respect, this is a proper generalization of convexity of the Lagrangian. We also specify a copositivity-based variant which is sufficient for global optimality. For (non-convex) quadratic optimization problems over polyhedra (QPs), the distinction between sufficiency and necessity vanishes, both for local and global optimality. However, in the strictly copositive case we can provide a distance lower (error) bound of the increment around a local minimizer . This is a refinement of an earlier result which focussed on mere (non-strict) copositivity. In addition, an apparently new variant of constraint qualification (CQ) is presented which is implied by Abadie’s CQ and which is suitable for second-order analysis. This new reflected Abadie CQ is neither implied, nor implies, Guignard’s CQ. However, it implies the necessary second-order local optimality condition based on copositivity. Application to the trust-region problem and several (counter) examples illustrates the advantage of this approach.