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**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)**Lecture September 30, 2016**- Classification of optimization problems. A production planning problem (Ref. material of the 2nd lecture)**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.**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)**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)**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 )**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)**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)**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.**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).**Lecture November 3, 2016**- The KKT conditions for inequality and equality constraints. The optimality condition for LP. (Ref. Chapt 6 of Teaching Notes)**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)**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).**Lecture November 11, 2016**- Primal-dual relationships. Exercises.**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)**Lecture November 18, 2016**- Fundamental theorem of Linear Programming (Ref. material of the 15th lecture)**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.**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)**Lecture December 1, 2016**- Branch and Bound (Ref. material Chapter 10)**Lecture December 2, 2016**- Branch and Bound: exercises (Ref. material Chapter 10)**Lecture December 15, 2016**- Multiobjective optimization: Pareto optimality. An example. Exercise from a test exam