6 semester


Title of the discipline (basic disciplines)




Course of Study


3, Mathematics (economic activities)



Semester of training
(for each semester a separate table)




Amount of credits
(academic plan)




Full name of the lecturer


Bakhtin Victor Ivanovich


Objectives of studying the discipline


Increase the level of professional competence in solving problems of optimization in various areas of work. Expansion of the mathematical outlook, acquaintance with new methods of proof, assimilation of new algorithms for solving optimization problems




Algebra and number theory,

Discrete Math,

Mathematical analysis,

Differential equations,

The theory of functions of a complex variable,

Functional Analysis

Extremal problems and the calculus of variations


Contents of the discipline



Topic 1.1. The general optimization problem.

Theme 1.2. Finding the minima and maxima of functions for unconditional optimization problems in finite-dimensional spaces.

Section 2. The principle of Lagrange multipliers in finite-dimensional spaces.

Topic 2.1. General optimization problem with constraints.

Theme 2.2. The principle of Lagrange for problems with constraints such as equalities.

Topic 2.3.The principle of Lagrange for problems with constraints such as equalities and inequalities.

Subject 2.4.A sufficient condition for an extremum for problems with constraints of the equality type.

2.5. A sufficient condition for an extremum for problems with mixed restrictions.

Section 3. Linear Programming

Theme 3.1. The problem of linear programming. Geometric interpretation of the linear programming problem.

The theme 3.2. Convex sets, their properties. Separation theorems.

Topic 3.3.Black points in the canonical linear problem. Nondegenerate problems. Simplex method.

Theme 3.4 The duality theory.

Section 4. Convex optimization problems

The theme 4.1. Convex functions. The problem of convex programming.

Theme 4.2. The optimality condition in the problem of convex programming.

Topic 4.3.The Slater condition and the Kuhn-Tucker optimality criterion


Recommended literature



1. Alekseev VM, Galeev EM, Tikhomirov VM Collection of problems on optimization. Theory. Examples. Tasks: Textbook. – Moscow: Science, 1984.

2. Gabasov R., Kirillova F. M. Optimization methods. 2nd edition. – Minsk: BSU Publishing House, 1981.

3. Galeev E.M. Optimization. Theory. Examples. Tasks. – Moscow: ComBook, 2006.

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

5. Gorokhovik V.V. Finite-dimensional optimization problems. – Minsk: 2006.


Teaching Methods



interactive methods of teaching (working in small groups (team), problem training) are organized taking into account the inclusion in the learning process of all students of the group. Joint activity means that each student makes his own individual contribution, in the course of the work there is an exchange of knowledge, ideas, methods of activity. Organized individual, steam and group work. Interactive methods are based on the principles of interaction, activity of trainees, reliance on group experience, mandatory feedback


Language of instruction







Conditions (requirements), routine monitoring



– test;
– Colloquium



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