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Journal of Convex Analysis 30 (2023), No. 2, 541--589
Copyright Heldermann Verlag 2023

Why Second-Order Sufficient Conditions are, in a Way, Easy -- or -- Revisiting the Calculus for Second Subderivatives

Matús Benko
Institut für Mathematik, Applied Mathematics and Optimization, University of Vienna, Austria

Patrick Mehlitz
Institute of Mathematics, Brandenburgische Technische Universität, Cottbus-Senftenberg, Germany

We readdress the classical topic of second-order sufficient optimality conditions for optimization problems with nonsmooth structure. Based on the so-called second subderivative of the objective function and of the indicator function associated with the feasible set, one easily obtains second-order sufficient optimality conditions of abstract form. In order to exploit further structure of the problem, e.g., composite terms in the objective function or feasible sets given as (images of) pre-images of closed sets under smooth transformations, to make these conditions fully explicit, we study calculus rules for the second subderivative under mild conditions. To be precise, we investigate a chain rule and a marginal function rule, which then also give a pre-image and image rule, respectively. As it turns out, the chain rule and the pre-image rule yield lower estimates, desirable in order to obtain sufficient optimality conditions, for free. Similar estimates for the marginal function and the image rule are valid under a comparatively mild inner calmness* assumption. Our findings are illustrated by several examples including problems from composite, disjunctive, and nonlinear second-order cone programming.

Keywords: Composite optimization, optimization with geometric constraints, second-order sufficient optimality conditions, second-order variational calculus, second subderivative.

MSC: 49J52, 49J53, 90C26, 90C46.

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