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**Lecture September 23, 2019**- Introduction to the course, exams and grading, teaching material. Online Survey. A brief review on OR history.**MATERIAL:**Slide 1st lecture, Chap1-2 of Hillier, Lieberman - Introduction to Operations Research, McGraw-Hill Education (2015), Dantzig's memory, George Dantzig in the development of economic analysis (by k. J. Arrow, Discrete Optimization, 5 (2), 2008)**Lecture September 27, 2019**- Paradigm for construction of mathematical models. An assignment problem. A simple production planning problem**MATERIAL:**slide 2nd lecture. description of the production problem.**Lecture September 30, 2019**- Basic definition and classification of optimization problems The model of the production planning problem.**MATERIAL:**teaching notes Chapter 1;**Lecture October 1, 2019**- Convex analysis: convex sets (definition and properties) and convex functions (definition). Convex optimization problem: definitions. Solution of the simple production problem.**MATERIAL:**teaching notes Chapter 2; Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed.**Lecture October 4, 2019**- Convex analysis: convex functions: First and second order conditions for convexity of a function.**MATERIAL:**chapter 2 Teaching Notes, Appendix B of D. Bertsekas, Nonlinear Programming- 2nd ed.). A multiplant optimization problem.**MATERIAL:**description of the multiplant problem.**Lecture October 7, 2019**- Theorem of equivalence of local and global minimizers with proof.**Lecture October 8, 2019**- Criteria for checking positive (semi)definiteness of a matrix. Convex and strictly convex quadratic functions: convexity criteria.**Lecture October 11, 2019**- Evaluation test in the class (text, survey results). The railway revenue management problem (description).**Lecture October 14, 2019**- Concave optimization problem: definition and Thorem of non existence of interior solution (with proof). Quadratic functions**Lecture October 15, 2019**-**Lecture October 18, 2019**- NO LECTURE**Lecture October 21, 2019**- Feasible and descent directions. Definitions. An example (**MATERIAL:**chapter 3 Teaching Notes, )**Lecture October 22, 2019**- Feasible directions: characterization for different feasible set. Unconstrained case; convex set;polyhedral set.**Lecture October 25, 2019**- Feasible directions with linear inequality constraints: characterizing the feasible directions and the maximum stepsize. Exercise derived from the text of Exam Februray 4, 2019**Lecture October 28, 2019**- Descent direction: first and second order charactherization; an example: finding a feasible and descent direction from the text of Exam Februray 4, 2019**Lecture October 29, 2019**- Optimality condition. Principles of feasible and descent algorithms (stopping criteria, main iteration).**Lecture November 4, 2019 -**Optimality conditions for the unconstrained case (stationary points) and for the convex constrained case.**Lecture November 5, 2019 -**Unconstrained optimality conditions using second order information. The gradient method (MATERIAL: Chapter 3 (pp236-238, 257-259) of D. Bertsekas, Nonlinear Programming- 3rd ed.). The conditional gradient method**Lecture November 11, 2019 - E**xamples of applications of the gradient methods and conditional gradient method (MATERIAL: slide of the lecture). The optimality conditions in the case of linear equality constraints**Lecture November 12, 2019 -**Optimization with linear equalities: the Lagrangian conditions (MATERIAL: (ref. Chapt 5 of Teaching Notes or Example 4.1.2 of Chapter 4 of D. Bertsekas, Nonlinear Programming- 3rd ed.)**Lecture November 18, 2019 -**Optimization with linear inequalities. The Farkas Lemma (Ref. Chapt 6 of Teaching Notes). The LP model of a transportation problem. The model of a lot sizing.**Lecture November 18, 2019 -**The KKT conditions for inequality linear constraints. A numerical example. Modelling fixed cost problem**Lecture November 25, 2019 -**OPIS Questionarries in the class. The optimality condition for LP. (Ref. Chapt 6 of Teaching Notes). Evaluation test in the class (text with solution)**Lecture November 26, 2019 -**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 December 2, 2019 -**Weak duality theorem (no proof).**Lecture December 3, 2019 -**The Dual problem of a blending problem (MATERIAL slide of the lecture). Sensitivity analysis: variation of the r.h.s. of the constraints**Lecture December 6, 2019 -**The continuous knapsack problem: finding the otimal solution using duality. (Ref. material Chapter 10). Extreme point of a convex set**Lecture December 9, 2019 -**Definition of vertex of a polyhedron and theorem on the characterization (no proof) - Fundamental Theorem of LP (no proof)**Lecture December 10, 2019 -**The standard form of LP. Basic feasible solutions and vertex. The reduced problem and the reduced cots**Lecture December 11, 2019 -**(A6 17:00-19:00) Basic of the simplex method.Integer Linear Programming: basic concept. Integer polyhedron, total unimodularity (no characterization).

Basic feasible solutions -**Lecture December 7, 2018 -**- 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**- 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)