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1. |
Title of the course |
Elements of calculus of variations and methods of optimization |
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2. |
Year and speciality |
3, Mathematics (scientific and design activity) |
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3. |
Semester |
5 |
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4. |
Number of credits
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3 |
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5. |
Lecturer |
Bakhtin Victor Ivanovich |
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6. |
Objectives of the course |
Acquaintance with basic methods for solution of the classical problems of finite-dimensional 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 finite-dimensional domains; — to construct models of extremal problems in finite-dimensional 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 simplex-method for linear programs; — to apply the first and second order optimality conditions for problems of nonlinear programming.
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7. |
Prerequisites |
Calculus, linear algebra
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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. Simplex-method. The duality theory. Problems of convex programming. Optimality conditions in convex problems. The condition of Slater and the Kuhn–Tucker theorem.
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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. Finite-dimensional 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).
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10. |
Teaching methods |
Lectures, exercises, supervised independent work |
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11. |
Language |
Russian |
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12. |
Conditions (requirements), running control |
Tests |
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13. |
Form of assessment |
Test |