There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.

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Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods

Background * Inequality constraints * Fenchel duality * Convex analysis * Special cases * Nonsmooth optimization * The Karush-Kuhn-Tucker Theorem * Fixed points * Postscript: infinite versus finite dimensions * List of results and notation.

Convex analysis and nonlinear optimization : theory and examples

http://static.luiss.it/hey/ambiguity/papers/Ghirardato_Maccheroni_Marinacci_2004.pdf

Differentiating ambiguity and ambiguity attitude

A Model Problem.- Semiconcave Functions.- Generalized Gradients and Semiconcavity.- Singularities of Semiconcave Functions.- Hamilton-Jacobi Equations.- Calculus of Variations.- Optimal Control Problems.- Control Problems with Exit Time.

/pdf/semiconcave-functions-hamilton-jacobi-equations-and-optimal-1vuumifya9.pdf

Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C This is shown to be equivalent to the prox-regularity of C at x, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation Additional characterizations are provided in terms of dC being locally of class C 1+ or such that dC + σ| · |2 is convex around x for some σ > 0 Prox-regularity of C at x corresponds further to the normal cone mapping NC having a hypomonotone truncation around x, and leads to a formula for PC by way of NC  The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting

/pdf/local-differentiability-of-distance-functions-4aduviqlr8.pdf

Local differentiability of distance functions

Prox-regularity of a set (Poliquin-Rockafellar-Thibault, 2000), or its global version, proximal smoothness (Clarke-Stern-Wolenski, 1995) plays an important role in variational analysis, not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist (C;.), or the local uniqueness of the projection mapping, but also because in the case where C is the epigraph of a locally Lipschitz function, it is equivalent to the weak convexity (lower-C 2 property) of the function. In this paper we provide an adapted geometrical concept, called sub smoothness, which permits an epigraphic characterization of the approximate convex functions (or lower-C 1 property). Subsmooth sets turn out to be naturally situated between the classes of prox-regular and of nearly radial sets. This latter class has been recently introduced by Lewis in 2002. We hereby relate it to the Mifflin semismooth functions.

https://www.ams.org/journals/tran/2005-357-04/S0002-9947-04-03718-3/S0002-9947-04-03718-3.pdf

Subsmooth sets: Functional characterizations and related concepts

On generalized differentials and subdifferentials of Lipschitz vector-valued functions

Prox-regular functions in Hilbert spaces

In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R

 n
 . In this context, we establish a subdifferential characterization and show that qualified convexly C
1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C
1 in the context of certain uniformly convex spaces.

Lionel Thibault

Papers

Local differentiability of distance functions

Subsmooth sets: Functional characterizations and related concepts

On generalized differentials and subdifferentials of Lipschitz vector-valued functions

Prox-regular functions in Hilbert spaces

Prox-Regularity of Functions and Sets in Banach Spaces