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Variational Convexity and the Local Monotonicity of Subgradient Mappings

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Abstract

It is well known that the subgradient mapping associated with a lower semicontinuous function is maximal monotone if and only if the function is convex, but what characterization can be given for the case in which a subgradient mapping is only maximal monotone locally instead of globally? That question is answered here in terms of a condition more subtle than local convexity. Applications are made to the tilt stability of a local minimum and to the local execution of the proximal point algorithm in optimization.

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Notes

  1. In fact, it is maximal cyclically monotone, since such monotonicity characterizes the subgradient mappings of lsc convex functions [16, 12.25].

  2. In a more recent article than here, we have shown that this condition can be weakened (see [15, Theorem 2]).

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

    R. Tyrrell Rockafellar

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  1. R. Tyrrell Rockafellar
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Correspondence to R. Tyrrell Rockafellar.

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This article is dedicated to Alex Ioffe in appreciation of our long-time friendship and mathematical partnership.

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Rockafellar, R.T. Variational Convexity and the Local Monotonicity of Subgradient Mappings. Vietnam J. Math. 47, 547–561 (2019). https://doi.org/10.1007/s10013-019-00339-5

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