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Name of the discipline  

EXTREME OBJECTIVES AND VARIATIONAL CALCULUS

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Study course,

specialty

3, Mathematics (economic activity)

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Semester of study

5

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Amount of credits     

4

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Lecturer’s S.N.P.

Gorokhovik Valentin Vikentievich

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Aims of the study

Increase the level of professional competence in solving problems of optimization in various areas of work. Expansion of the mathematical horizon, familiarity with new methods of proof, mastering of new algorithms for solving optimization problems.

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.

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Prerequisites

Mathematical analysis

Algebra and Number Theory

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Contents of the discipline

The general optimization problem. Finding the minima and maxima of functions for problems of unconditional optimization in finite-dimensional spaces. The principle of Lagrange multipliers in finite-dimensional spaces. General optimization problem with constraints. The Lagrange principle for problems with constraints of the type of equations. The Lagrange principle for problems with constraints such as equalities and inequalities. A sufficient condition for an extremum for problems with constraints such as equalities. A sufficient condition for an extremum for problems with mixed constraints. Linear programming. The problem of linear programming. Geometric interpretation of the linear programming problem. Convex sets, their properties. Separation theorems. The extreme points in the canonical linear problem. Nondegenerate problems. Simplex method. The duality theory. Convex optimization problems. Convex functions. The problem of convex programming. Optimality conditions in the problem of convex programming. The Slater condition and the Kuhn-Tucker optimality criterion.

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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.

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Methods of teaching

Lectures, laboratory practice, SSI (students scientific investigation)

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Language of teaching

Russian

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Knowledge control and requirements

Tests

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Certification form

Passed, exam. The score on the exam is set taking into account the current academic performance (with a coefficient of 0.3)