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Optimization Theory - practicals (WS 2024/2025)
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Conditions for the pass:
- 1 homework (will be given in the middle of the semester)
- 1 written test: 70% points are necessary, the test can be repeated only once
- Date of the test: TBD.
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Literature:
- COLLECTION OF EXAMPLES with solutions (will be updated during the semester): HERE.
- Nonlinear programming and optimality conditions: VIDEO.
- Lecture notes by doc. Lachout available HERE .
- Sample test: TBA.
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Additional literature/sources:
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- Video files from 2021 ZOOM exercises in MOODLE.
- Bazaraa, M.S., Sherali, H.D., and Shetty, C.M. (2006). Nonlinear programming: theory and algorithms, Wiley, Singapore, 3rd edition.
- Old lecture notes by doc. Lachout here (IN CZECH)
- Dupačová, J., Lachout, P. (2011). Úvod do optimalizace, Matfyzpress, Praha. (IN CZECH)
- Boyd, S., Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press, Cambridge.
- Rockafellar, R.T. (1972). Convex analysis, Princeton University Press, New Jersey.
- Rockafellar, R.T., Wets, R. (2004). Variational analysis, Springer-Verlag, Berlin, 2nd edition.
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Practicals 1
- Introduction to optimization, formulations of optimization problems.
- Examples: Section 1
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Practicals 2
- Extremes of functions, convex sets I
- Examples: Section 2 (Ex. 2.1, 2.3), Section 3 (Ex. 3.2)
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Practicals 3
- Convex sets II
- Examples: Ex. 3.4, 3.6, 3.7 (2, 3), 3.10 (1, 2, 4)
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Practicals 4
- Separating hyperplane theorems
- Examples: 4.1, 4.2, 4.4, 4.6
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Practicals 5
- Farkas theorem
- Examples: 4.8, 4.9, 4.10
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Practicals 6
- Linear programming I (graphical solution, direct approach)
- Examples: 5.1, 5.2, 5.3
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Practicals 7
- Linear programming II (duality)
- Examples: 5.5, 5.6
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Practicals 8
- Linear programming III (duality, simplex algorithm)
- Examples: 5.8
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Practicals 9
- Linear programming IV (simplex algorithm)
- Examples: 5.9
- Convex functions I
- Examples: 6.2, 6.4 (2, 3)
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Practicals 10
- Convex functions II
- Examples: 6.4 (5, 7), 6.6 (2, 6), 6.13 (1, 4)
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Practicals 11
- Nonlinear optimization (optimality conditions)
- Examples: 7.1, 7.3, 7.13
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Practicals 12
- TEST
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Optimization Theory - practicals (WS 2021/2022)
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Conditions for the pass:
- 2 written tests (70% points from each test are necessary); each test can be repeated only once at the end of the semester.
- If the classroom teaching is cancelled, the course will be taught by distant education methods.
- Expected date of the first test: 24 and 26 November 2021.
- Expected date of the second test: 5 and 7 January 2022.
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Literature:
- COLLECTION OF EXAMPLES with solutions (can be extended during the semester): HERE.
- Lecture notes by doc. Lachout available HERE .
- Video files from ZOOM exercises in MOODLE.
- Sample test: HERE.
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Additional literature:
-
- Bazaraa, M.S., Sherali, H.D., and Shetty, C.M. (2006). Nonlinear programming: theory and algorithms, Wiley, Singapore, 3rd edition.
- Skripta doc. Lachouta volně ke stažení ZDE (IN CZECH)
- Dupačová, J., Lachout, P. (2011). Úvod do optimalizace, Matfyzpress, Praha. (IN CZECH)
- Boyd, S., Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press, Cambridge.
- Rockafellar, R.T. (1972). Convex analysis, Princeton University Press, New Jersey.
- Rockafellar, R.T., Wets, R. (2004). Variational analysis, Springer-Verlag, Berlin, 2nd edition.
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Practicals 1
- Introduction to optimization: linear programming (duality, structure of the feasibility set), nonlinear programming (SLPO, constraints qualification).
- Examples: 1.2, 1.3
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Practicals 2
- Projection of a point to a convex set, separating and supporting hyperplanes.
- Examples: 2.1, 2.2
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Practicals 3
- Separation of two convex sets.
- Examples: 2.4, 2.5, 2.6, 2.8
- VIDEO.
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Practicals 4
- Convexity of sets and functions I.
- Examples: 3.1, 3.2, 3.3, 3.4 (2, 3, 7, 9), 3.5 (12), 3.6 (6, 11, 13)
- VIDEO.
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Practicals 5
- Convexity of sets and functions II. Subdifferentiability.
- Examples: 3.8, 3.9, 4.1, 4.3, 4.4
- VIDEO.
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Practicals 6
- Quasiconvex functions I.
- Examples: 5.1, 5.2, 5.3, 5.4, 5.5, 5.8
- VIDEO.
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Practicals 7
- Quasiconvex functions II. Pseudoconvex functions.
- Examples: 5.10 (1, 4), 5.11, 5.12, 5.14, 5.15, 5.16, 5.18, 5.20, 5.21
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Practicals 8
- Test 1.
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Practicals 9
- Optimality conditions based on directions.
- Examples: 6.1, 6.2, 6.3, 6.4, 6.6, 6.7
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Practicals 10
- Karush-Kuhn-Tucker optimality conditions I.
- Examples: 6.9, 6.13, 6.19
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Practicals 11
- Karush-Kuhn-Tucker optimality conditions II (CQ, SOSC).
- Examples: 6.15, 6.17, 6.23, 6.24, 6.27
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Practicals 12
- Karush-Kuhn-Tucker optimality conditions III (CQ, SOSC).
- Examples: 6.26, 6.28, 6.25
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Practicals 13
- Test 2.