|SCIENTIFIC DISCIPLINARY SECTOR||MAT/09|
The course provides skills related to the construction of models and the solution of decision-making problems formulated as optimization problems. Moreover, the course presents the main methods of statistics for the description of data and the extraction of information from them.
The aim of this course is to provide the tools which allow, on the one hand, to characterize phenomena and physical systems from a statistical point of view by using measured data and, on the other hand, to formulate and solve optimization problems with continuous, integer or mixed variables, also in the presence of constraints.
The course aims to study the main optimization methods for solving decision-making problems and basic techniques from Statistics to describe a phenomenon and generate knowledge from data.
In more detail, as regards the optimization part, the course aims to provide students with the basic skills for the mathematical formalization and the subsequent solution of decision-making problems, in which it is necessary to take optimal decisions within many possible ones, based on suitable criteria. In particular, the course presents the concepts of decision variables, objective function, and constraints of an optimization problem, as well as the basic notions of real linear programming, integer linear programming, and nonlinear programming.
As regards the statistics part, the course provides basic notions of descriptive statistics and inferential statistics, in order to allow the students to appropriately describe a set of data coming from observations of a quantity of interest, as well as to extract information from data, that is, build a model of what is observed starting from a limited set of observations.
In both cases, both methodological and applicative aspects are presented. The various concepts are exposed through theoretical lessons and through the solution of exercises, as well as through the software implementation of some example problems.
At the end of the course, the students will be able to construct a mathematical model of a decision-making process and to choose and apply the most appropriate algorithm for its solution. Furthermore, the students will be able to describe the data collected in the field and extract information from them through the most appropriate tools.
Basic knowledge of Calculus.
PART OF OPTIMIZATION
1. Introduction to the optimization part
1.1. Definitions of objective function and decision variables
1.2. Classification of optimization problems
1.3. Mathematical formulation of optimization problems
2. Real linear programming
2.1. Examples of linear programming problems
2.2. Geometry of linear programming
2.3. Standard form of a linear programming problem
2.4. Simplex algorithm
2.5. Two-phase method
3. Integer linear programming
3.1. Examples of integer linear programming problems
3.2. Branch and bound method
4. Nonlinear programming
4.1. Examples of nonlinear programming problems
4.2. Necessary conditions of optimality and sufficient conditions of optimality
4.3. Descent algorithms
4.4. Gradient algorithm
4.5. Lagrangian approach for constrained problems
4.6. Penalty functions method
5. Software for mathematical programming
5.1. General introduction to Lingo, Matlab, Excel
5.2. Examples of use of software for solving linear and nonlinear programming problems
PART OF STATISTICS
6. Introduction to the statistics part
6.1. Preliminary definitions
7. Descriptive statistics
7.1. Qualitative variables
7.2. Graphical representation
7.3. Quantitative variables
7.4. Mode, median, mean, percentiles, quartiles
7.5. Matlab software for data representation
8. Regression and least squares
8.1. Bivariate data samples
8.2. Linear regression
8.3. Least squares for function approximation
8.4. Matlab software for linear regression and least squares
9. Review of the theory of probability
9.1. Combinatorial calculus
9.2. The concept of probability
9.3. Random variables
9.4. Probability density
9.5. The most common probability distributions
10. Estimates, estimators, confidence intervals
10.1. Point and interval estimates
10.2. Confidence intervals
Handouts provided by the lecturer in electroic format..
Books for further readings:
 Hillier, Lieberman – Introduction to operations research. McGraw-Hill, 2004.
 D. Bertsimas, J.N. Tsitsiklis – Introduction to linear optimization. Athena Scientific, 1999.
 D. Luenberger, Y. Ye – Linear and nonlinear programming. Springer, 2008.
 D. Bertsekas – Nonlinear Programming. Athena Scientific, 1999.
 S.M. Ross – Probabilità e statistica per l’ingegneria e le scienze, Apogeo, 2014.
 P. Newbold, W.L. Carlson, B. Thorne – Statistica, Pearson, 2010.
Office hours: By appointment to be requested by email.
As reported in the official calendar.
All class schedules are posted on the EasyAcademy portal.
Written examination possibly supplemented by oral examination.
At the end of the course, the students will have to demonstrate that they have understood the concepts discussed in class and be able to explain them using an appropriate language. Furthermore, the students will have to demonstrate the ability to constract a mathematical model of a decision-making process and to choose and apply the best algorithm for its solution. Finally, the students must be able to describe a set of data collected in the field and to extract information from them through the most appropriate tools.