Copositivity for second-order optimality conditions in general smooth optimization problems. (2nd April 2016)

• Record Type: Journal Article
• Authors: Bomze, Immanuel M.
• Is Part Of: Optimization. Volume 65:Number 4(2016)
• Page Start: 779
• Page End: 795
• Publication Date: 2016-04-02
• Subjects: copositive matrices -- non-convex optimization -- global optimality condition -- polynomial optimization -- trust region problem; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2015.1083019
• Languages: English
• ISSNs: 0233-1934

Abstract

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.