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Asymptotic Equivalence of Evolution Equations Governed by Cocoercive Operators and Their Forward Discretizations

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Abstract

The purpose of this work is to study discrete approximations of evolution equations governed by cocoercive operators by means of Euler iterations, both in a finite and in an infinite time horizon. On the one hand, we give precise estimations for the distance between iterates of independently generated Euler sequences and use them to obtain bounds for the distance between the state, given by the continuous-time trajectory, and the discrete approximation obtained by the Euler iterations. On the other hand, we establish the asymptotic equivalence between the continuous- and discrete-time systems, under a sharp hypothesis on the step sizes, which can be removed for operators deriving from a potential. As a consequence, we are able to construct a family of smooth functions for which the trajectories/sequences generated by basic first-order methods converge weakly but not strongly, extending the counterexample of Baillon. Finally, we include a few guidelines to address the problem in smooth Banach spaces.

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Notes

  1. Or m-accretive in the Banach space terminology.

  2. This is a piecewise constant interpolation of the sequence \(\mathcal E_nx\).

  3. A different extension would be to consider \(B:X\rightarrow X^*\), to cover the case \(B=\nabla f\). We shall not explore this path here.

  4. The arguments also hold for q-cocoercive operators in q-uniformly smooth spaces (see, for instance, [39]).

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Acknowledgements

Supported by Fondecyt Grant 1181179 and Basal Project CMM Universidad de Chile. Andrés Contreras was also supported by CONICYT-PCHA/Doctorado Nacional/2016-21160994.

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Authors and Affiliations

  1. Departamento de Ingeniería Matemática, FCFM, Universidad de Chile, Beauchef 851, Santiago, Chile

    Andrés Contreras

  2. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (CNRS UMI2807), FCFM, Universidad de Chile, Beauchef 851, Santiago, Chile

    Juan Peypouquet

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  1. Andrés Contreras
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  2. Juan Peypouquet
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Correspondence to Juan Peypouquet.

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Communicated by Michel Théra.

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Contreras, A., Peypouquet, J. Asymptotic Equivalence of Evolution Equations Governed by Cocoercive Operators and Their Forward Discretizations. J Optim Theory Appl 182, 30–48 (2019). https://doi.org/10.1007/s10957-018-1332-3

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