## Lectures 2019

1. Lecture September 23, 2019 - Introduction to the course, exams and grading, teaching material. Online Survey.  A brief review on OR history.   MATERIAL: Slide 1st lecture, Chap1-2 of Hillier,  Lieberman - Introduction to Operations Research, McGraw-Hill Education (2015), Dantzig's memory, George Dantzig in the development of economic analysis (by  k. J. Arrow, Discrete Optimization, 5 (2), 2008)
2. Lecture September 27, 2019 - Paradigm for  construction of mathematical models.  An assignment problem. A simple production planning problem MATERIAL: slide 2nd lecture. description of the production problem.
3. Lecture September 30, 2019 - Basic definition and classification of optimization problems The model of the production planning problem. MATERIAL: teaching notes Chapter 1;
4. Lecture October 1, 2019 - Convex analysis: convex sets (definition and properties) and convex functions (definition). Convex optimization problem: definitions. Solution of the simple production problem. MATERIAL: teaching notes Chapter 2; Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed.
5. Lecture October 4, 2019 - Convex analysis: convex functions: First and second order conditions for convexity of a function. MATERIAL:  chapter 2 Teaching Notes, Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed.). A multiplant optimization problem. MATERIAL: description of the multiplant problem.
6. Lecture October 7, 2019 -  Theorem of equivalence of local  and global minimizers with proof.
7. Lecture October 8, 2019 - Criteria for checking positive (semi)definiteness of a matrix. Convex and strictly convex quadratic functions: convexity criteria.
8. Lecture October 11, 2019 -  Evaluation test in the class (text, survey results). The railway revenue management problem (description).
9. Lecture October 14, 2019 - Concave optimization problem: definition and Thorem of non existence of interior solution (with proof). Quadratic functions
10. Lecture October 15, 2019 -
11. Lecture October 18, 2019 - NO LECTURE
12. Lecture October 21, 2019 - Feasible and descent directions. Definitions. An example (MATERIAL:  chapter 3 Teaching Notes, )
13. Lecture October 22, 2019 - Feasible directions: characterization for different feasible set. Unconstrained case; convex set;polyhedral set.
14. Lecture October 25, 2019 - Feasible directions with linear inequality constraints: characterizing the feasible directions and the maximum stepsize. Exercise derived from the text of Exam Februray 4, 2019
15. Lecture October 28, 2019 - Descent direction: first and second order charactherization; an example: finding a feasible and descent direction from the text of Exam Februray 4, 2019
16. Lecture October 29, 2019 -  Optimality condition. Principles of feasible and descent algorithms (stopping criteria, main iteration).
17. Lecture November 4, 2019 - Optimality conditions for the unconstrained case (stationary points) and for the convex constrained case.
18. Lecture November 5, 2019 - Unconstrained optimality conditions using second order information. The gradient method (MATERIAL: Chapter 3 (pp236-238, 257-259) of D. Bertsekas, Nonlinear Programming- 3rd ed.). The conditional gradient method
19. Lecture November 11, 2019 - Examples of applications of the gradient methods and conditional gradient method (MATERIAL: slide of the lecture). The optimality conditions in the case of linear equality constraints
20. Lecture November 12, 2019 -  Optimization with linear equalities: the Lagrangian conditions (MATERIAL:  (ref. Chapt 5 of Teaching Notes or Example 4.1.2 of Chapter 4 of D. Bertsekas, Nonlinear Programming- 3rd ed.)
21. Lecture November 18, 2019 - Optimization with linear inequalities. The Farkas Lemma (Ref. Chapt 6 of Teaching Notes). The LP model of a transportation problem. The model of a lot sizing
22. Lecture November 18, 2019 -  The KKT conditions for inequality linear constraints. A numerical example. Modelling fixed cost problem
23. Lecture November 25, 2019 -  OPIS Questionarries in the class. The optimality condition for LP. (Ref. Chapt 6 of Teaching Notes). Evaluation test in the class (text with solution)
24. Lecture November 26, 2019 - Duality for LP: strong duality theorem (Ref. Chapt 7 of Teaching Notes and Chapt 4 Kwon, Roy H. Introduction to linear optimization and extensions with MATLAB, CRC Press (2014)).
25. Lecture December 2, 2019 - Weak duality theorem (no proof).
26. Lecture December 3, 2019 - The Dual problem of a blending problem (MATERIAL slide of the lecture). Sensitivity analysis: variation of the r.h.s. of the constraints
27. Lecture December 6, 2019 - The continuous knapsack problem: finding the otimal solution using duality. (Ref. material Chapter 10). Extreme point of a convex set
28. Lecture December 9, 2019 - Definition of vertex of a polyhedron and theorem on the characterization (no proof) - Fundamental Theorem of LP (no proof)
29. Lecture December 10, 2019 - The standard form of LP. Basic feasible solutions and vertex. The reduced problem and the reduced cots
30. Lecture December 11, 2019 - (A6 17:00-19:00) Basic of the simplex method.Integer Linear Programming: basic concept. Integer polyhedron, total unimodularity (no characterization).
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3. Lecture December 7, 2018 - Basic feasible solutions -
4. Lecture December 13, 2018 -  - Upepr and lower bound - Solution of the continuous knapsack problem. The dual of the blending problem, the dual of the transportation problem.
5. Lecture - Integer linear Programming: definition and first examples. The continuous knapsack problem: finding the otimal solution using duality. (Ref. material Chapter 10)
6. Lecture  Branch and Bound (Ref. material Chapter 10)
7. Lecture - Branch and Bound: exercises (Ref. material Chapter 10)