Samir ADLY : The variational convexity module: spectral curvature invariant and exact transport via Moreau regularization
In this work, in collaboration with Prof. R.T. Rockafellar, we present the variational convexity module associated with a proximal basis pair. This module provides a quantitative measure of the local curvature of a function that may be non-smooth and non-convex. It describes the best quadratic convexity or hypoconvexity behavior compatible with the variational geometry of the subdifferential. It can thus be viewed as a local conditioning parameter: it plays a role in the stability of critical points, in the proper definition of proximal or resolvent maps, and in the local analysis of proximal splitting methods and first-order methods beyond the smooth setting.
We then recall a recent fundamental result by Rockafellar: this module admits an exact spectral characterization using a second-order object called the strict second subdifferential. In other words, it can be interpreted as the smallest local curvature observed in the admissible directions. This result also provides a precise criterion for prox-regularity: a function is prox-regular exactly when this module is not equal to negative infinity.
Based on this spectral interpretation, we explain how this module transforms under proximal regularization, and more specifically under the Moreau envelope. We highlight a mechanism for switching between the strict second derivative and Moreau regularization
Based on this spectral interpretation, we explain how this module transforms under proximal regularization, and more specifically under the Moreau envelope. We highlight a switching mechanism between the strict second derivative and Moreau regularization, which leads to an explicit curvature attenuation law.
From this, we derive stability results for the variational convexity module of the Moreau envelope, as well as an inversion formula that allows us to recover the curvature of the initial function from its regularized version. The talk concludes with several examples—smooth, nonsmooth, and truly non-convex—that illustrate the cases where this spectral interpretation is fully relevant and those where it degenerates.
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