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On implicit functions in nonsmooth analysis

Author

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  • DORSCH, Dominik

    () (RWTH Aachen University, Germany)

  • JONGEN, Hubertus Th.

    () (RWTH Aachen University , Germany)

  • RÜCKMANN, Jan.-J.

    () (School of Mathematics, The University of Birmingham, U.K.)

  • SHIKHMAN, Vladimir

    () (Université catholique de Louvain, CORE, Belgium)

Abstract

We study systems of equations, F (x) = 0, given by piecewise differentiable functions F : Rn → Rk, k ≤ n. The focus is on the representability of the solution set locally as an (n − k)-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarke’s generalized Jacobians. The SFRA claims the existence of a basis in which Clarke’s inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarke’s generalized Jacobian. The article is devoted to the conjectured equivalence of SFRA and FRA. For min-type functions, we give reformulations of SFRA and FRA using orthogonal projections, basis enlargements, cross products, dual variables, as well as via exponentially many convex cones. The equivalence of SFRA and FRA is shown to be true for min-type functions in the new case k = 3.

Suggested Citation

  • DORSCH, Dominik & JONGEN, Hubertus Th. & RÜCKMANN, Jan.-J. & SHIKHMAN, Vladimir, 2013. "On implicit functions in nonsmooth analysis," CORE Discussion Papers 2013021, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvco:2013021
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    References listed on IDEAS

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    1. Fleurbaey,Marc & Maniquet,François, 2011. "A Theory of Fairness and Social Welfare," Cambridge Books, Cambridge University Press, number 9780521715348, May.
    2. Duranton, Gilles & Martin, Philippe & Mayer, Thierry & Mayneris, Florian, 2010. "The Economics of Clusters: Lessons from the French Experience," OUP Catalogue, Oxford University Press, number 9780199592203.
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    Keywords

    Clarke's inverse function theorem; strong full-rank assumption; full-rank assumption; full-rank conjecture; Lipschitz manifold;

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