7 semester

1.       

Name of the discipline  

 VARIATIONAL CALCULATION AND METHODS OF OPTIMIZATION

2.       

Study course,

specialty

4, Mathematics and Information Technology (mathematical and software of mobile devices)

3.       

Semester of study

7

4.       

Amount of credits                     

3

5.       

Lecturer’s S.N.P.

Lebedev Andrey Vladimirovich

6.       

Aims of the study

The study of the basic methods for solving classical problems of finite-dimensional optimization. Increase the level of professional competence in solving optimization problems. Further development of students’ skills in abstract mathematical thinking and the ability to apply it in specific tasks, increase their mathematical culture.

As a result of the study, the student should be able to:

– find the minimum and maximum points for functions defined on finite-dimensional spaces;

– construct models of extremal problems in finite-dimensional spaces;

– with the aid of differential convexity criteria, to check whether a given function is convex;

– use optimality conditions and the Kuhn-Tucker criterion for solving convex programming problems;

– use the simplex method for solving linear programming problems;

– use the optimality conditions of the first and second order to solve nonlinear programming problems.

7.       

Prerequisites

Mathematical analysis

Algebra and Number Theory

Differential equations

Optimization methods in finite-dimensional spaces

Functional Analysis

8.       

Contents of the discipline

Optimization problems in infinite-dimensional spaces. The problem of brachistochrone.

Introduction to the calculus of variations. History and significance in the development of infinite-dimensional analysis.

The simplest variational problem.

Strong and weak extrema in the simplest variational problem

Variations of the objective functional of the simplest variational problem.

Differentiation of functions and mappings defined on normed spaces. Definitions of the first and second variations of Lagrange. Derivatives of Gato and Frechet.

 General view of the first and second variations of the integral functional.

Necessary conditions for a local minimum in the simplest variational problem.

Conditions for a local minimum of first and second order for functions defined on normed spaces. Singularities of sufficient conditions for a local minimum in infinite-dimensional spaces. Necessary conditions of the first and second order for the local minimum of the simplest variational problem in terms of variations of the target functional.

Integral and differential Euler equations. The Weierstrass-Erdman condition. Hilbert’s theorem. Theory of the second variation. A necessary condition for Legendre. Necessary condition for Jacobi.

A sufficient condition for a weak local minimum in the simplest variational problem.

The use of a sufficient Jacobi condition for a weak local minimum in the simplest variational problems.

A necessary Weierstrass condition for a strong minimum in the simplest variational problem.

Isoperimetric variational problem.

The local minimum in the isoperimetric variational problem.

9.       

Recommended literature

1. Galeev EM, Tikhomirov VM A short course in the theory of extremal problems. – Moscow, Moscow State University, 1989.

2. Alekseev VM, Galeev EM, Tikhomirov VM Collection of tasks on optimization. Theory. Examples, Tasks. Study Guide: – Moscow, Science, 1984.

3. Gorokhovik V.V. Finite-dimensional optimization problems. – Minsk, 2006.

4. Bakhtin VI, Ivanishko IA, Lebedev AV, Pindrik OI Linear programming. Method. Allowance

5. Bakhtin VI, Ivanishko IA, Lebedev AV, Pindrik OI The Lagrange principle. Method. allowance

10.   

Methods of teaching

Lectures, laboratory practice, SSI (students scientific investigation)

11.   

Language of teaching

Russian

12.   

Knowledge control and requirements

test (when examining an examination the current academic performance is taken into account with a coefficient of 0.3)

13.   

Certification form

Exam