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Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization

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Abstract

Many problems in nonlinear analysis and optimization, among them variational inequalities and minimization of convex functions, can be reduced to finding zeros (namely, roots) of set-valued operators. Hence numerous algorithms have been devised in order to achieve this task. A lot of these algorithms are inexact in the sense that they allow perturbations to appear during the iterative process, and hence they enable one to better deal with noise and computational errors, as well as superiorization. For many years a certain fundamental question has remained open regarding many of these known inexact algorithmic schemes in various finite and infinite dimensional settings, namely whether there exist sequences satisfying these inexact schemes when errors appear. We provide a positive answer to this question. Our results also show that various theorems discussing the convergence of these inexact schemes have a genuine merit beyond the exact case. As a by-product we solve the standard and the strongly implicit inexact resolvent inclusion problems, introduce a promising class of functions (fully Legendre functions), establish continuous dependence (stability) properties of the solution of the inexact resolvent inclusion problem and continuity properties of the protoresolvent, and generalize the notion of strong monotonicity.

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Acknowledgements

We would like to express our thanks to Shoham Sabach and Roman Polyak for helpful discussions, and to the referees for considering our paper and for their feedback. Part of the research of the first author was done while he was at the Institute of Mathematical and Computer Sciences (ICMC), University of São Paulo, São Carlos, Brazil (2015) and this is an opportunity for him to thank FAPESP. The second author was partially supported by the Israel Science Foundation (Grant 389/12), by the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.

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    Daniel Reem & Simeon Reich

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Reem, D., Reich, S. Solutions to inexact resolvent inclusion problems with applications to nonlinear analysis and optimization. Rend. Circ. Mat. Palermo, II. Ser 67, 337–371 (2018). https://doi.org/10.1007/s12215-017-0318-6

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