4 semester


Name of the discipline



Course of Study



Semester of training

4, mathematics (economic activity)


Amount of credits



Full name of lecturer

Gorokhovik Valentin Vikentievich


Objectives of studying the discipline

the formation of the fundamentals of mathematical thinking among students, familiarity with the methods of mathematical proofs, the construction of geometric models using the concept of convexity and analytical models and problems in which convex functions and concepts close to them participate, as well as the study of algorithms for solving specific mathematical problems associated with the notion of convexity.

As a result of studying the discipline, the trainee must:

be able to:

– find solutions to problems on convex sets and convex functions;

– independently to orient in the literature on the subject convex sets and convex functions.



Mathematical analysis


Contents of the discipline

Definition of a convex set, examples. Topological properties of convex sets. Affine spaces and affine shells. Determination of the projection of a point onto a set. Separability of convex sets. Properties of the relative interior of a convex set. The cones: definitions and the simplest properties. Extreme points: definitions and examples. Asymptotic cones and the Klee theorem. Combinatorial properties of convex sets. Convex polyhedra: definitions and examples. Multifaceted cones and their properties. Separation theorems for cones. Brauer’s theorem. Combinatorial lemma on convex means. Convex functions. Convex functions: definition and examples. Criteria of convexity. A theorem on the continuity and Lipschitz property of a convex function. The differentiability of convex functions. Closed functions. Conjugated functions. The subdifferential. The Moro-RocaFellar theorem and its applications.


Recommended literature

1. Danzer A., ​​Grünbaum B., Klee V. The Helly theorem and its applications. – Moscow: The World, 1968.

2. Zabreiko PP Convex sets. – Minsk: Belarusian State University, 1984.

3. Ioffe AD, Tikhomirov VM The theory of extremal problems. – Moscow: Nauka, Home Edition of Physical and Mathematical Literature, 1974.

4. Magaril – Ilyaev GG, Tikhomirov VM Convex analysis and its applications. – Moscow: Editorial URSS, 2000.

5. Pshenichny BN Convex analysis and extremal problems. – Moscow: Science, the main edition of physics and mathematics, 1980.

6. Roberts A.W., Varberg D.E. Convex functions. – Academic Press, 1973.

7. Rokafellar R. Convex analysis. – Moscow: The World, 1973.

8. Stoer J., Witzgall C. Convexity and optimization in finite dimensional. I. – Springer – Verlag, Berlin, 1970.


Teaching Methods

Lectures, practical exercises, UIRS


Language of education



Conditions (requirements), current control

– colloquium;


Form of certification