• ## Optimization Theory - practicals (WS 2023/2024)

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: 11 January 2024.

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.

• 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.
• Practicals 1
Introduction to optimization, formulations of optimization problems.
Examples: Section 1
Practicals 2
Extremes of functions, convex sets I
Examples: Section 2 (Ex. 2.1, 2.3), Section 3 (Ex. 3.2)
Practicals 3
Convex sets II
Examples: Ex. 3.4, 3.6, 3.7 (2, 3), 3.10 (1, 2, 4)
Practicals 4
Separating hyperplane theorems
Examples: 4.1, 4.2, 4.4, 4.6
Practicals 5
Farkas theorem
Examples: 4.8, 4.9, 4.10
Practicals 6
Linear programming I (graphical solution, direct approach)
Examples: 5.1, 5.2, 5.3
Practicals 7
Linear programming II (duality)
Examples: 5.5, 5.6
Practicals 8
Linear programming III (duality, simplex algorithm)
Examples: 5.8
Practicals 9
Linear programming IV (simplex algorithm)
Examples: 5.9
Convex functions I
Examples: 6.2, 6.4 (2, 3)
Practicals 10
Convex functions II
Examples: 6.4 (5, 7), 6.6 (2, 6), 6.13 (1, 4)
Practicals 11
Nonlinear optimization (optimality conditions)
Examples: 7.1, 7.3, 7.13
Practicals 12
TEST
• ## Optimization Theory - practicals (WS 2021/2022)

-
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.

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.

• 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.
• Practicals 1
Introduction to optimization: linear programming (duality, structure of the feasibility set), nonlinear programming (SLPO, constraints qualification).
Examples: 1.2, 1.3
Practicals 2
Projection of a point to a convex set, separating and supporting hyperplanes.
Examples: 2.1, 2.2
Practicals 3
Separation of two convex sets.
Examples: 2.4, 2.5, 2.6, 2.8
VIDEO.
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.
Practicals 5
Convexity of sets and functions II. Subdifferentiability.
Examples: 3.8, 3.9, 4.1, 4.3, 4.4
VIDEO.
Practicals 6
Quasiconvex functions I.
Examples: 5.1, 5.2, 5.3, 5.4, 5.5, 5.8
VIDEO.
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
Practicals 8
Test 1.
Practicals 9
Optimality conditions based on directions.
Examples: 6.1, 6.2, 6.3, 6.4, 6.6, 6.7
Practicals 10
Karush-Kuhn-Tucker optimality conditions I.
Examples: 6.9, 6.13, 6.19
Practicals 11
Karush-Kuhn-Tucker optimality conditions II (CQ, SOSC).
Examples: 6.15, 6.17, 6.23, 6.24, 6.27
Practicals 12
Karush-Kuhn-Tucker optimality conditions III (CQ, SOSC).
Examples: 6.26, 6.28, 6.25
Practicals 13
Test 2.