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**Lecture September 25, 2017**- Introduction to the course, exams and grading, teaching material. A brief review on OR history. Paradigm for construction of mathematical models. An assignment problem. (Ref. material of the 1st lecture)**Lecture September 27, 2017**- An investment optimization model. A production planning problem (Ref. material of the 2nd lecture)**Lecture October 2, 2017**- Basic defintion and classification of optimization problems A nonlilnear model of optimal sizing.**Lecture October 4, 2017**- Convex analysis: convex sets (definition and properties) and convex functions (definition). Convex optimization problem: definition (Ref. Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed.). A multiplant optimization problem and graphical solution.**Lecture October 6, 2017**- Convex optimization problem: definition and theorem of equivalence of local and global minimizers (Ref. Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed., chapter 2 Teaching Notes)**Lecture October 9, 2017**- Concave optimization problem: defintion and non existence of interior solution. Quadratic functions. Convex and strictly convex quadratic functions: convexity criteria. (Ref. chapter 2 Teaching Notes)**Lecture October 11, 2017**- Criteria for checking positive (semi)definiteness of a matrix. Quadratic functions. Exercise from exam tests of June 27, 2016 and July 19, 2017, June 15, 2017. (Ref material of the 4th lecture)**Lecture October 13, 2017**- Descent and feasible directions.**Lecture October 16, 2017**- First and second order characterization of descent directions. The case on unconstrained problem: first order necessary conditions.**Lecture October 18, 2017**- Second order necessary conditionsfor unconstrained optimization. The convex uconstrained case.**Lecture October 20, 2017**-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)**Lecture October 23, 2017**- Feasible direction of a polyhedron. Exercise from text exams.**Lecture October 25, 2016**- Optimization over a polyhedron: feasible directions, maximum stepsize for feasiblity, first order conditions.**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.)**Lecture November 6, 2017**- Optimization with inequality: Farkas' Lemma and the KKT conditions. (Ref. Chapt 5 of Teaching Notes).**Lecture November 8, 2017**- The KKT conditions for inequality and equality constraints. The optimality condition for LP. (Ref. Chapt 6 of Teaching Notes)**Lecture November 10, 2017**- Duality for LP: weak 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)).**Lecture November 13, 2017**- 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)).**Lecture November 15, 2017**- Construction of priml-dual LP problems. Use of the KKT (duality) theorem to find pair of primal-dual solutions. Exercise from the exams.**Lecture November 17, 2017**- Modelling absolute values in min function (min max).**Lecture November 20, 2017**- Sensitivity analysis in LP**Lecture November 22, 2017**- The dual of a blending problem.**Lecture November 24, 2017**- Primal-dual relationships. The continuous knapsack example. Exercises.A blending model with advertising**Lecture November 27, 2017**- 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 29, 2017**- Fundamental theorem of Linear Programming (Ref. material of the 15th lecture)**Lecture December 1, 2017**- 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 December 11, 2017**- Integer linear Programming: definition and first examples. The continuous knapsack problem: finding the otimal solution using duality. (Ref. material Chapter 10)**Lecture December 13, 2017**- Branch and Bound (Ref. material Chapter 10)**Lecture December 15, 2017**- Branch and Bound: exercises (Ref. material Chapter 10)**Lecture December 18, 2017**- Multiobjective optimization: Pareto optimality. An example. Exercise from a test exam