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Name of the discipline 
EXTREME OBJECTIVES AND VARIATIONAL CALCULUS

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Study course, specialty 
3, Mathematics (scientific and production 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 finitedimensional spaces; – construct models of extremal problems in finitedimensional spaces; – with the aid of differential convexity criteria, to check whether a given function is convex; – use optimality conditions and the KuhnTucker 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 finitedimensional spaces. The principle of Lagrange multipliers in finitedimensional 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 KuhnTucker 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. Finitedimensional 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) 