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**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,**Lecture September 28, 2018**- A simple production planning problem**MATERIAL:**slide 2nd lecture, description of the production problem.**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;**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.**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)**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.**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.**Lecture October 19, 2018**- First and second order characterization of descent directions. The case on unconstrained problem: first or necessary conditions.**Lecture October 25, 2018 -**Second order necessary conditionsfor unconstrained optimization. The convex uconstrained case. Exercise**No****Lecture October 26, 2018****No Lecture November 1, 2018****No Lecture November 2, 2018****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)**Lecture November 9, 2018**- Feasible direction of a polyhedron.**Lecture November 15, 2018**- Optimization over a polyhedron: feasible directions, maximum stepsize for feasiblity, first order conditions.**Lecture November 16, 2018 -****Lecture November****Lecture November****Lecture November****Lecture November**Definition of vertex of a polyhedron and theorem on the characterization (no proof) - Fundamental Theorem of LP (no proof)**Lecture December 6, 2018 -**Basic feasible solutions - Basic of the simplex method**Lecture December 7, 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.**Lecture December 13, 2018 -****Lecture December 14, 2018****Lecture December 15, 2018**

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**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 5, 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 10, 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 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.**Lecture November 15, 2017**- Modelling absolute values in min function (min max).**Lecture November 17, 2017**- Sensitivity analysis in LP**Lecture November 20, 2017**- The dual of a blending problem.**Lecture November 22, 2017**- Primal-dual relationships. The continuous knapsack example. Exercises.A blending model with advertising- - Extreme point of convex sets. Vertex of a polyhedron: characterization theorem. Exercises. A simple lot sizing model. (Ref. material of the 15th lecture)
**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.**Lecture**- Integer linear Programming: definition and first examples. The continuous knapsack problem: finding the otimal solution using duality. (Ref. material Chapter 10)**Lecture**Branch and Bound (Ref. material Chapter 10)**Lecture**- Branch and Bound: exercises (Ref. material Chapter 10)**Lecture**- Multiobjective optimization: Pareto optimality. An example. Exercise from a test exam