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On smoothness properties of optimal value functions at the boundary of their domain under complete convexity

Oliver Stein (stein***at***kit.edu)
Nathan Sudermann-Merx (sudermann***at***kit.edu)

Abstract: This article studies continuity and directional differentiability properties of optimal value functions, in particular at boundary points of their domain. We extend and complement standard continuity results from W.W. Hogan, Point-to-set maps in mathematical programming, SIAM Review, Vol. 15 (1973), 591-603, for abstract feasible set mappings under complete convexity as well as standard differentiability results from W.W. Hogan, Directional derivatives for extremal-value functions with applications to the completely convex case, Operations Research, Vol. 21 (1973), 188-209, for feasible set mappings in functional form under the Slater condition in the unfolded feasible set. In particular, we present sufficient conditions for the inner semi-continuity of feasible set mappings and, using techniques from nonsmooth analysis, provide functional descriptions of tangent cones to the domain of the optimal value function. The latter makes the stated directional differentiability results accessible for practical applications.

Keywords: Complete convexity, Slater condition, inner semi-continuity, directional differentiability, nonsmooth linearization cone

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: Mathematical Methods of Operations Research, DOI 10.1007/s00186-014-0465-x

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Entry Submitted: 06/14/2013
Entry Accepted: 06/16/2013
Entry Last Modified: 03/10/2014

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