Subdifferentials of convex functions

Random variables can be described by their cumulative distribution functions, a class of nondecreasing functions on the real line. Those functions can in turn be identified, after the possible vertical gaps in their graphs are filled in, with maximal monotone relations. Such relations are known to be the subdifferentials of convex functions. Analysis of these connections yields new insights. The generalized inversion operation between distribution functions and quantile functions corresponds to graphical inversion of monotone relations. In subdifferential terms, it corresponds to passing to conjugate convex functions under the Legendre---Fenchel transform. Among other things, this shows that convergence in distribution for sequences of random variables is equivalent to graphical convergence of the monotone relations and epigraphical convergence of the associated convex functions. Measures of risk that employ quantiles (VaR) and superquantiles (CVaR), either individually or in mixtures, are illuminated in this way. Formulas for their calculation are seen from a perspective that reveals how they were discovered. The approach leads further to developments in which the superquantiles for a given distribution are interpreted as the quantiles for an overlying "superdistribution." In this way a generalization of Koenker---Basset error is derived which lays a foundation for superquantile regression as a higher-order extension of quantile regression.

The crucial discovery of the concept of differential calculus is due to Pierre de Fermat (1601–1655), who was one of the most important innovators in the history of mathematics. It is to him that we owe a rule for determining extrema, described, without proof, in a short treatise Methodus ad disquirendam Maximum et Minimam written in 1637. The importance of his discoveries in number theory has eclipsed the contributions which this exceptional and modest man made to other areas of mathematics. Fermat also was the first to discover the “principle of least time” in optics, the prototype of the variational principles governing so many physical and mechanical laws. He shared independently with Descartes the invention of analytic geometry and with Pascal the creation of the mathematical theory of probability. His achievements in number theory overshadowed his other contributions, as the Last Fermat Theorem which remained a challenge for such a long time, and still is a challenge if indeed the simple proof of Piere de Fermat did exist. Not to mention his compositions in French, Latin, Italian and Spanish verse and his Grecian erudition. It is also notable that he was able to find time for these occupations in the midst of his duties as counsellor to the parliament of Toulouse (even taking into account Fermat’s genius, this makes us reflect on the leisure activities offered by a lawyer’s career).

In this chapter, controlled differential inclusions in a Hilbert space containing subdifferentials of convex functions are considered. The following three problems are studied: the problem of prescribed motion realization, the problem of robust control, and the problem of input dynamical reconstruction. Solution algorithms that are stable with respect to informational noises and computational errors are presented. The algorithms are based on the method of feedback control. They adaptively take into account inaccurate measurements of phase trajectories and are regularized in the following sense: the more precise is incoming information, the better is the algorithm’s output.

Subdifferentiability of Convex Functions with Values in an Ordered Vector Space.

Variational formulations of non-linear and non-smooth structural problems

On the convergence of subdifferentials of convex functions

In the paper we deal with the problem when the graph of the subdifferential operator of a convex lower semicontinuous function has a common point with the product of two convex nonempty weak and weak* compact sets, i.e. when graph ∂ψ ∩ (Q × Q*) ≠ 0. The results obtained partially solve the problem posed by Simons as well as generalize the Rockafellar Maximal Monotonicity Theorem.