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Sensitivity analysis of maximally monotone inclusions via the proto-differentiability of the resolvent operator

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Abstract

This paper is devoted to the study of sensitivity to perturbation of parametrized variational inclusions involving maximally monotone operators in a Hilbert space. The perturbation of all the data involved in the problem is taken into account. Using the concept of proto-differentiability of a multifunction and the notion of semi-differentiability of a single-valued map, we establish the differentiability of the solution of a parametrized monotone inclusion. We also give an exact formula of the proto-derivative of the resolvent operator associated to the maximally monotone parameterized variational inclusion. This shows that the derivative of the solution of the parametrized variational inclusion obeys the same pattern by being itself a solution of a variational inclusion involving the semi-derivative and the proto-derivative of the associated maps. An application to the study of the sensitivity analysis of a parametrized primal-dual composite monotone inclusion is given. Under some sufficient conditions on the data, it is shown that the primal and the dual solutions are differentiable and their derivatives belong to the derivative of the associated Kuhn–Tucker set.

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Acknowledgements

This research benefited from the support of the FMJH “Program Gaspard Monge for optimization and operations research and their interactions with data science”, and from the support from EDF, Thales and Orange

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  1. Laboratoire XLIM, Université de Limoges, 123 Avenue Albert Thomas, Limoges CEDEX, 87060, France

    Samir Adly

  2. Department of Mathematics, University of Washington, Box 354350, Seattle, WA, 98195-4350, USA

    R. Tyrrell Rockafellar

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  1. Samir Adly
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  2. R. Tyrrell Rockafellar
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Correspondence to Samir Adly.

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Adly, S., Rockafellar, R.T. Sensitivity analysis of maximally monotone inclusions via the proto-differentiability of the resolvent operator. Math. Program. 189, 37–54 (2021). https://doi.org/10.1007/s10107-020-01515-z

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