Forthcoming seminars
Seminar calendar
List of seminars
Archive by years
Register a seminar

Forthcoming seminars

You may need the following programs to see the files

Steklov Mathematical Institute Seminar
April 16, 2015 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

Degenerate problems of nonlinear analysis and extremum theory

A. V. Arutyunov
Video records:
MP4 1,654.0 Mb
MP4 419.6 Mb

Number of views:
This page:869
Video files:256
Youtube Video:

A. V. Arutyunov
Photo Gallery

Видео не загружается в Ваш браузер:
  1. Установите Adobe Flash Player    

  2. Проверьте с Вашим администратором, что из Вашей сети разрешены исходящие соединения на порт 8080
  3. Сообщите администратору портала о данной ошибке

Abstract: Consider a system of nonlinear equations $\mathbf F(x)=y$, where $\mathbf F$ is a smooth map of some Banach space $X$ into another Banach space (for simplicity, these spaces can be regarded as finite-dimensional). If the point $x_0$ is degenerate, that is, the linear operator $\mathbf F'(x_0)$ is not surjective (e.g., $\mathbf F'(x_0)=0$), then, at the point $x_0$, the classical inverse function theorem can not be applied. Herein, we present theorems on inverse and implicit functions which can be applied to degenerate points as well.
Consider the classical extremal problem with constraints:
$$ \varphi(x)\to\min,\qquad f_i(x)=0,\quad i=1,2,…,k,\qquad x\in X. $$

Here, the smooth functions fi define the constraints, and $\varphi$ is the minimizing functional. Let $x_0$ be a local minimum. If the point $x_0$ is degenerate (abnormal), that is, the gradients $f_i'(x_0)$ are linearly dependent, then the Lagrange principle degenerates (i.e., it becomes non-informative), and the classical second order necessary optimality conditions do not hold true. We present a theory of necessary conditions of the first and second order which is equally meaningful for degenerate and non-degenerate problems. These results are a further development of the Lagrange principle.
A classic example of such abnormal problem: would a given quadratic form be non-negative (or, would it be zero) at the intersection of quadrics? The present theory allows us to give answers to these questions.
All the outlined in the report results are meaningful in the finite-dimensional case as well (even when $X$ is 3-dimensional).

  1. A. V. Arutyunov, “Gladkie anormalnye zadachi teorii ekstremuma i analiza”, UMN, 67:3(405) (2012), 3–62  mathnet  crossref  mathscinet  zmath  adsnasa; A. V. Arutyunov, “Smooth abnormal problems in extremum theory and analysis”, Russian Math. Surveys, 67:3 (2012), 403–457  crossref  mathscinet  zmath  isi  scopus

SHARE: FaceBook Twitter Livejournal
Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2017