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

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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 Hogan (SIAM Rev 15:591–603, 1973a) for abstract feasible set mappings under complete convexity as well as standard differentiability results from Hogan (Oper Res 21:188–209, 1973b) 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.

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Acknowledgments

We thank the anonymous referee for his or her precise and substantial remarks which helped to significantly improve the paper. Moreover, we are grateful to Christian Kanzow, Diethard Klatte and Robert Mohr for fruitful discussions on earlier versions of this paper, and to Georg Still for pointing out the reference Schneider (1993) to us.

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Authors and Affiliations

  1. Institute of Operations Research, Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany

    Oliver Stein & Nathan Sudermann-Merx

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  1. Oliver Stein
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  2. Nathan Sudermann-Merx
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Correspondence to Oliver Stein.

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This research was partially supported by the DFG (Deutsche Forschungsgemeinschaft) under grant STE 772/13-1.

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Stein, O., Sudermann-Merx, N. On smoothness properties of optimal value functions at the boundary of their domain under complete convexity. Math Meth Oper Res 79, 327–352 (2014). https://doi.org/10.1007/s00186-014-0465-x

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