## Lectures 2018

1. Lecture September 27, 2018 - Introduction to the course, exams and grading, teaching material. Online Survey.  A brief review on OR history. Paradigm for  construction of mathematical models.  An assignment problem.  MATERIAL: Slide 1st lecture, Chap1-2 of Hillier,  Lieberman - Introduction to Operations Research, McGraw-Hill Education (2015), Dantzig's memory,
2. Lecture September 28, 2018 - A simple production planning problem MATERIAL: slide 2nd lecture, description of the production problem.
3. Lecture October 4, 2018 - Basic defintion and classification of optimization problems A nonlilnear model of optimal sizing. MATERIAL: teaching notes Chapter 1; description of the optimal sizing problem;
4. Lecture October 5, 2018 - Convex analysis: convex sets (definition and properties) and convex functions (definition). Convex optimization problem: definitions. A multiplant optimization problem. MATERIAL: teaching notes Chapter 2; Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed., description of the multiplant problem.
5. Lecture October 11, 2018 - Convex optimization problem: theorem of equivalence of local  and global minimizers. First and second order conditions for convexity of a function. (Ref. Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed., chapter 2 Teaching Notes)
6. Lecture October 12, 2018 -  Criteria for checking positive (semi)definiteness of a matrix. Convex and strictly convex quadratic functions: convexity criteria. Evaluation test in the class.
7. Lecture October 18, 2018 - Concave optimization problem: defintion and non existence of interior solution. Quadratic functions.   (Ref. chapter 2 Teaching Notes) -Descent and feasible directions.
8. Lecture October 19, 2018 First and second order characterization of descent directions. The case on unconstrained problem: first or necessary conditions.
9. Lecture October 25, 2018 - Second order necessary conditionsfor unconstrained optimization. The convex uconstrained case. Exercise
10. No Lecture October 26, 2018
11. No Lecture November 1, 2018
12. No Lecture November 2, 2018
13. Lecture November 8, 2018 - The gradient method. A production model from the text exam. (Ref. Chapter 3 (pp236-238, 257-259) of D. Bertsekas, Nonlinear Programming- 3rd ed. - material of the lecture)
14. Lecture November 9, 2018 - Feasible direction of a polyhedron.
15. Lecture November 15, 2018 - Optimization over a polyhedron: feasible directions, maximum stepsize for feasiblity, first order conditions.
16. Lecture November 16, 2018 -
17. Lecture November
18. Lecture November
19. Lecture November
20. Lecture November
21. Lecture December 6, 2018 - Definition of vertex of a polyhedron and theorem on the characterization (no proof) - Fundamental Theorem of LP (no proof)
22. Lecture December 7, 2018 - Basic feasible solutions - Basic of the simplex method
23. Lecture December 13, 2018 - Integer Linear Programming: basic concept. Integer polyhedron, total unimodularity (no characterization). - Upepr and lower bound - Solution of the continuous knapsack problem. The dual of the blending problem, the dual of the transportation problem.
24. Lecture December 14, 2018
25. Lecture December 15, 2018

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1. Lecture October 30, 2017  Optimization with linear equality: the Lagrangian conditions.  (ref. Chapt 5 of Teaching Notes or Example 4.1.2 of Chapter 4 of D. Bertsekas, Nonlinear Programming- 3rd ed.)
2. Lecture November 5, 2017 Optimization with inequality: Farkas' Lemma and the KKT conditions.  (Ref. Chapt 5 of Teaching Notes).
3. Lecture November 8, 2017 - The KKT conditions for inequality and equality constraints. The optimality condition for LP. (Ref. Chapt 6 of Teaching Notes)
4. Lecture November 10, 2017 - Duality for LP: weak  duality theorem (Ref. Chapt 7 of Teaching Notes.
5. Lecture November 10, 2017 - Duality for LP: strong duality theorem (Ref. Chapt 7 of Teaching Notes
6. Lecture November 13, 2017 - Construction of priml-dual LP problems. Use of the KKT (duality) theorem to find pair of primal-dual solutions. Exercise from the exams.
7. Lecture November 15, 2017 -  Modelling absolute values in min function (min max).
8. Lecture November 17, 2017 - Sensitivity analysis in LP
9. Lecture November 20, 2017 - The dual of a blending problem.
10. Lecture November 22, 2017 - Primal-dual relationships. The continuous knapsack example. Exercises.A blending model with advertising
11. - Extreme point of convex sets. Vertex of a polyhedron: characterization theorem. Exercises. A simple lot sizing model. (Ref. material of the 15th lecture)
12.  - Fundamental theorem of Linear Programming (Ref. material of the 15th lecture)
13. Lecture  - The standard form of LP. Basic feasible solutions and vertex. The reduced problem and the reduced cots. A blending model with advertsing and logical constraints.
14. Lecture - Integer linear Programming: definition and first examples. The continuous knapsack problem: finding the otimal solution using duality. (Ref. material Chapter 10)
15. Lecture  Branch and Bound (Ref. material Chapter 10)
16. Lecture - Branch and Bound: exercises (Ref. material Chapter 10)
17. Lecture  - Multiobjective optimization: Pareto optimality. An example. Exercise from a test exam