Abstract
The objective of this short note is to provide an estimate of the generalized Jacobian of the inverse of a Lipschitzian mapping when Clarke’s inverse function theorem applies. Contrary to the classical \(\mathcal {C}^{1}\) case, inverting matrices of the generalized Jacobian is not enough. Simple counterexamples show that our results are sharp.
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References
Bartl, D., Fabian, M.: Every compact convex subset of marices is the Clarke Jacobian of some lipschitzian mapping. Proc. Am. Math. Soc. 149(11), 4771–4779 (2021)
Bolte, J., Le, T., Pauwels, E., Silvetto-Falls, A.: Nonsmooth implicit differentiation for machine learning and optimization. Advances in neural information processing systems, vol. 34 (2021)
Clarke, F.H.: On the inverse function theorem. Pac. J. Math. 64 (1), 97–102 (1976)
Fabian, M., Preiss, D.: On the Clarke’s generalized Jacobian. Suppl. Rend. Circ. Mat. Palermo 14, 305–307 (1987)
Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4(1), 79–97 (1979)
Outrata, J., Kocv̌ara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems With Equilibrium Constraints. Theory, Applications and Numerical Results. Nonconvex Optimization and its Application Series Vol. 28. Kluwer, Boston (1998)
Warga, J.: Fat homeomorphisms and unbounded derivative containers. J. Math. Anal. Appl. 81(2), 545–560 (1982). Erratum in J. Math. Anal. Appl. 82(2)582–583
Winston, E., Zico Kolter, J.: Monotone operator equilibrium networks. Advances in neural information processing systems, vol. 33 (2020)
Acknowledgments
The authors thanks D. Bartl and J. Outrata, as well as the two referees, for their comments which contributed to improving the overall presentation. M. Fabian’s work was supported by the grant of CACR 20 − 22230L and by RVO:679858840. E. Pauwels acknowledges the support of AI Interdisciplinary Institute ANITI funding, through the French “Investing for the Future - PIA3” program under the Grant agreement ANR-19-PI3A0004, Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA9550-19-1-7026, FA9550-18-1-0226, and ANR MaSDOL 19-CE23-0017-01.
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Fabian, M., Hiriart-Urruty, JB. & Pauwels, E. On the Generalized Jacobian of the Inverse of a Lipschitzian Mapping. Set-Valued Var. Anal 30, 1443–1451 (2022). https://doi.org/10.1007/s11228-022-00640-5
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DOI: https://doi.org/10.1007/s11228-022-00640-5