Lectures

  1. Lecture September 29, 2016 - Introduction to the course, exams and grading, teaching material. A brief review on OR history. Paradigm for  construction of mathematical models.  An assignment problem. An investment optimization model. (Ref. material of the 1st lecture)
  2. Lecture September 30, 2016 - Classification of optimization problems. A production planning problem (Ref. material of the 2nd lecture)
  3. Lecture October 6, 2016 - Convex analysis: convex sets (definition and properties) and convex functions (definition and characterization). Convex optimization problem: definition and theorem of equivalence of local  and global minimizers (Ref. Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed.) Exercises on convex problem from exam tests of June 27, 2016 and July 1, 2014. A nonlilnear model of optimal sizing.
  4. Lecture October 7, 2016 - Concave optimization problem: defintion and non existence of interior solution. Criteria for checking positive (semi)definiteness of a matrix. Quadratic functions. Exercise from exam tests of September 21, 2015 and January 25, 2016. (Ref material of the 4th lecture)
  5. Lecture October 13, 2016- Descent and feasible directions. First order characterization of descent directions. The case on unconstrained problem: first order necessary conditions. A multiplant optimization problem and graphical solution. Exercise from exam tests of November 3, 2014. (Ref.  material of the 5th lecture)
  6. Lecture October 14, 2016 - Second order characterization of descent directions. The case on unconstrained problem: second order necessary conditions, second order sufficient condition. Exercise from exam tests of February 20, 2014 and November 3, 2014. (Ref. Chapt 3 Teaching Notes )
  7. Lecture October 20, 2016 - A general scheme of unconstrained algorithms (the gradient method with exact linesearch). Optimization over a convex set (first and second order conditions). A railway Revenue Management problem. Exercise from exam tests of June 27, 2016. (Ref. Chapter 3 (pp236-238, 257-259) of D. Bertsekas, Nonlinear Programming- 3rd ed. - material of the 7th lecture) 
  8. Lecture October 21, 2016 -  Optimization over a polyhedron: feasible directions, maximum stepsize for feasiblity, first order conditions. Exercise from exam tests of June 27, 2016. (ref. Chapt 5 of Teaching Notes) 
  9. Lecture October 27, 2016 -  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.)   A production model from exam tests of June 27, 2016.
  10. Lecture October 28, 2016 - Optimization with inequality: Farkas' Lemma and the KKT conditions. Exercise from exam tests of July 25, 2016. (Ref. Chapt 5 of Teaching Notes).
  11. Lecture November 3, 2016 - The KKT conditions for inequality and equality constraints. The optimality condition for LP. (Ref. Chapt 6 of Teaching Notes)
  12. Lecture November 4, 2016 - Duality for LP: weak and strong duality theorems (Ref. Chapt 7 of Teaching Notes and Chapt 4 Kwon, Roy H. Introduction to linear optimization and extensions with MATLAB, CRC Press (2014)). Modelling absolute values in min function (min max). (Ref. material of the 12th lecture)
  13. Lecture November 10, 2016 - Construction of dual LP problems. Use of the KKT (duality) theorem to find pair of primal-dual solutions. Exercise from the exams of November 8, 2016 and February 9, 2015. The dual of a blending problem (Ref. material of the 13th lecture).
  14. Lecture November 11, 2016 - Primal-dual relationships. Exercises.
  15. Lecture November 17, 2016 - Extreme point of convex sets. Vertex of a polyhedron: characterization theorem. Exercises. A simple lot sizing model. (Ref. material of the 15th lecture)
  16. Lecture November 18, 2016 - Fundamental theorem of Linear Programming (Ref. material of the 15th lecture)
  17. Lecture November 24, 2016 - 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.
  18. Lecture November 25, 2016 - Integer linear Programming: definition and first examples. The continuous knapsack problem: finding the otimal solution using duality. (Ref. material Chapter 10)
  19. Lecture December 1, 2016 - Branch and Bound (Ref. material Chapter 10)
  20. Lecture December 2, 2016 - Branch and Bound: exercises (Ref. material Chapter 10)
  21. Lecture December 15, 2016 - Multiobjective optimization: Pareto optimality. An example. Exercise from a test exam