Méthodes Numérique Dans Les Problèmes D’ Extrémum | B. Pchénitchny, Y. Daniline

Numerical methods for solving various problems of the mind have taken off in recent years, so much so that the corresponding bibliography contains hundreds of books. This interest is by no means accidental, it reflects the leading role that extremum problems play in applications. This book is specifically devoted to the efficient search for the minimum of a function whose variables are subject to constraints.

Let us point out at once that the requirements formulated with regard to new algorithms are not the same as they were ten or fifteen years ago, when each new calculation procedure for a particular minimization problem was met with intérêt.Il it is currently not enough to build an algorithm, it is also necessary to show that it trumps the known ones. So we have to compare the effectiveness of different algorithms, a problem that unfortunately cannot be solved so easily. Indeed, we only compare based on one criterion. However, there are more than one (for example, the accuracy of the result, the calculation time, the necessary memory occupation), and it happens that we are asked to evaluate an algorithm according to several rather contradictory criteria.

In choosing the algorithms to be examined in this book, the authors essentially started from the criterion of accuracy of the result and that of the speed of convergence of the iterative process. Even if we stay within this narrow framework, however, we cannot unambiguously order all the algorithms or indicate the best and worst of them. The fact is that estimates of the convergence rate are obtained for classes of problems, not for isolated problems, and an algorithm that is bad for a large class may prove effective for another, more restricted one. The calculator must therefore have a whole arsenal of algorithms to be able to cope with each proposed problem.

It is also necessary to take into account the way in which a high speed of convergence is achieved. In practice, even the calculation of first derivatives of a function is often difficult and that of second derivatives inextricable. The authors therefore insisted on algorithms that require the calculation of the [only] first derivatives or the values of the function.

The authors place themselves in finite dimension. Because, firstly, in automatic computation, the solution of one problem must be approached by the solution of another in finite dimension, and, secondly, most algorithms generalize quite easily to the minimization of functionals without undergoing any essential modifications. The authors have therefore resolved to limit themselves to the finite-dimensional case. The book has become even more accessible to the general public since most of the results require only knowledge of the foundations of mathematical analysis and linear algebra to be understood.

In order not to burden the presentation, references in the text itself are very rare (they are usually collected in the brief comments at the end of each chapter). Since the literature on the issues dealt with is too abundant, the bibliography is mainly reinforced only by articles and monographs that the authors have used directly.

The present book completely glosses over the methods of solving the vast and important class of poorly posed problems of excess, methods developed by A. Tikhonov and his school. The authors barely touch the resolution of these problems of optimal control. These problems and the corresponding solving techniques are studied from various points of view in the monograph by N. Moïse, Numerical methods in the theory of optimal systems.

 Traduit du russe par Irina Pêtrova

Un grand merci à Henri Leveque pour le scan original.