OPTIMIZATION PLACES AND FACES. A JOURNEY TO ENDOLANDIA

Abstract

It was smooth on paper, But ravines have been forgotten Where one should walk, where one should walk…

Leo Tolstoy

Preface

The necessity to solve optimization problems was one of the main incentives for the appearance and development of Mathematics which, in turn, provided the tools and devices for solving such problems. Initially, only particular problems were studied, and for solving each of them, a specific method was to be invented. With the rise of Mathematical Analysis, the main tool of which is the notion of gradient (derivative in the one-dimensional case) introduced by I. Newton and G. Leibnitz, it became possible to solve a wide class of optimization problems, described by differentiable (“smooth”) functions. Nonsmooth problems were persistently knocking at the door, and P.L.Chebyshev opened the way for them, formulating and solving (140 years ago) the problem of polynomials least deviating from zero (the famous Chebyshev polynomials). In doing this, he employed the existing tools of (“smooth”) Mathematical Analysis. P.L.Chebyshev is therefore considered the God-father of Nonsmooth Analysis.

However, it was quite clear that for solving a wider class of nonsmooth problems, specific tools were required. U.Dini introduced the notion of directional derivative. The family of directionally differentiable functions is much broader than that of smooth functions. For the study of this family of functions, Nonsmooth Analysis (NSA) came into being. NSA is an extension and generalization of the classical (smooth) Mathematical Analysis. A breakthrough in this area was made in 1962 when J.-J. Moreau introduced the notion of subdifferential for convex functions. Since then, NSA has become a wellestablished branch of modern mathematics, and Nondifferentiable Optimization (NDO) has developed into a rich source of methods and algorithms for solving a variety of optimization problems (both smooth and nonsmooth).

This work is dedicated to the 50th anniversary of the birth of Nonsmooth Analysis and Nondifferentiable Optimization and represents the author’s photo account on his journeys to a wonderful country called OPTIMIZATION, or, to be more precise, to that part of it, where Nonsmooth Analysis is a Governor – the country of Nondifferentiable Optimization (NDO-landia, or just ENDOLANDIA).

The narration is organized as follows. The author depicts some events (such as conferences, workshops, summer and winter schools etc.) which he was fortunate enough to attend. Due to the limits of space and time, the author could describe in detail only twelve of these events. Chapter 13 is a digest that touches upon other events that were originally planned to be included. Chapter 14 contains photographs of the participants of the forthcoming Conference on Constructive Nonsmooth Analysis (CNSA) to be held in St. Petersburg from 18th to 23rd of June, 2012. The photos in this Chapter were contributed (with small exceptions) by the participants themselves. The book contains more than 1150 photographs representing more than 700 persons working in the NSA, NDO are related areas.

An electronic version of the book was prepared by L.K.Korsikova. The index of Names was composed by G.Sh.Tamasyan and N.A.Tolochinskaya. Several photos were kindly provided by V.V.Karelin. The cover design is due to Yu.Yu.Pupysheva. The assistance of E.V.Artem’eva, J.-P. Crouseix, S.Dempe, Z.Dzalilova, J.-B.Hiriart-Urruty, I.V.Romanovski, D.Pallaschke, V.K.Shomesova, M.Thera and V.N.Ushakov in identifying the persons shown in the pictures was invaluable. I.Shablinskaya and I.Willis gave useful comments on the text of the Preface.

Thus, welcome to ENDOLANDIA!

April 1, 2012 V.F.Demyanov