Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems

Helmut Gfrerer (), Jane J. Ye () and Jinchuan Zhou ()
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Helmut Gfrerer: Institute of Computational Mathematics, Johannes Kepler University Linz, A-4040 Linz, Austria
Jane J. Ye: Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 2Y2, Canada
Jinchuan Zhou: Department of Statistics, School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, P.R. China

Mathematics of Operations Research, 2022, vol. 47, issue 3, 2344-2365

Abstract: In this paper, we study second-order optimality conditions for nonconvex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper, we propose two approaches for establishing second-order optimality conditions for the nonconvex case. In the first approach, we extend the concept of the support function so that it is applicable to general nonconvex set-constrained problems, whereas in the second approach, we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

Keywords: Primary: 90C26; secondary: 90C46; 49J53; second-order tangent sets; second-order optimality conditions; lower generalized support function; directional metric subregularity; directional normal cones; directional regular tangent cones; directional Robinson’s constraint qualification; directional nondegeneracy (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:3:p:2344-2365

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