1. 
Title of the course 
Elements of calculus of variations and methods of optimization 
2. 
Year and speciality 
3, Mathematics (scientific and design activity) 
3. 
Semester 
5 
4. 
Number of credits

3 
5. 
Lecturer 
Bakhtin Victor Ivanovich 
6. 
Objectives of the course 
Acquaintance with basic methods for solution of the classical problems of finitedimensional optimization. Enhancement of professional competence for solution of the optimization problems. Further formation of students’ skills of abstract mathematical reasoning and ability to apply it to specific problems, upgrade of the mathematical knowledge common level.
As a result of the course study a student have to be able: — to find maxima and minima for functions with finitedimensional domains; — to construct models of extremal problems in finitedimensional spaces; — to verify convexity of a given function by means of differential criteria; — to apply optimality conditions and the Kuhn–Tucker criterion to solve problems of convex programming; — to apply simplexmethod for linear programs; — to apply the first and second order optimality conditions for problems of nonlinear programming.

7. 
Prerequisites 
Calculus, linear algebra

8. 
Contents of the course 
Optimization problems under presence/absence of constraints. The Lagrange principle for optimization problems with constraints in the form of equalities. The Lagrange principle for optimization problems with constraints in the form of equalities and inequalities. Sufficient conditions for extrema in problems with constraints in the form of equalities (or both equalities and inequalities). The problem of linear programming. Convex sets, separation theorems. Extremum points in the canonical linear problem. Nondegenerate linear problems. Simplexmethod. The duality theory. Problems of convex programming. Optimality conditions in convex problems. The condition of Slater and the Kuhn–Tucker theorem.

9. 
Recommended literature 
1. Galeev, E.M.; Tikhomirov, V.M. A concise course of the extremal problems theory. – Moscow, MGU, 1989 (in Russian). 2. Alekseev, V.M.; Galeev, E.M.; Tikhomirov, V.M. A workbook on optimization. Theory. Examples. Problems. – Moscow, “Nauka”, 1984 (in Russian) 3. Gorokhovik, V.V. Finitedimensional optimization problems. – Minsk, 2006 (in Russian) 4. Bakhtin, V.I.; Ivanishko, I.A.; Lebedev, A.V.; Pindrik, O.I. Linear programming (learner’s guide). 5. Bakhtin, V.I.; Ivanishko, I.A.; Lebedev, A.V.; Pindrik, O.I. The Lagrange principle (learner’s guide).

10. 
Teaching methods 
Lectures, exercises, supervised independent work 
11. 
Language 
Russian 
12. 
Conditions (requirements), running control 
Tests 
13. 
Form of assessment 
Test 