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

Immanuel M. Bomze(immanuel.bomze***at***univie.ac.at)

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 (nonconvex) 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 examples illustrate the advantage of this approach.

Keywords: Copositive matrices, non-convex optimization, global optimality condition, polynomial optimization, trust region problem

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 2: Nonlinear Optimization (Quadratic Programming )

Citation: Preprint NI13033-POP, Isaac Newton Institute for Mathematical Sciences, University of Cambridge UK, submitted

Download: [PDF]

Entry Submitted: 02/19/2015
Entry Accepted: 02/19/2015
Entry Last Modified: 02/19/2015

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