Graph Theory, Discrete Mathematics and Optimization
A.Y. 2020/2021
Learning objectives
This course aims at introducing modern and advanced mathematical techniques useful for understanding and modeling big data structures and algorithms.
Expected learning outcomes
At the end of the course the student will be able to formalize real world problems in mathematical terms and to solve simple exercises related with linear algebra, graph theory, Markov Chains, Optimization and Decision Theory.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
The lessons will take place online in synchronous way on the platforms MSTeams or Zoom during the timetable fixed by the faculty. The lectures will also be registered and made available to the students to be followed in an asynchronous way.
The program of the course will remain unchanged.
The written exams during the emergency will maintain the same structure of the usual ones and will be organized through the platform Exam.net, with an external supervision by Zoom or MSTeams.
The program of the course will remain unchanged.
The written exams during the emergency will maintain the same structure of the usual ones and will be organized through the platform Exam.net, with an external supervision by Zoom or MSTeams.
Prerequisites for admission
Prerequisites for this course include a good knowledge of the mathematical tools presented in Calculus I and Linear Algebra courses and in an introductory course in Probability and Statistics, with particular reference to the following arguments: real functions of one and two variables; derivatives; partial derivatives; integrals; vectors and matrices; operations between matrices; determinants; eigenvalues and eigenvectors; probability measures; events; random variables; distributions.
Assessment methods and Criteria
he final examination consists of a written exam during which the student must solve some exercises in the format of open-ended and/or multiple answer questions, with the aim of assessing the student's knowledge and ability to solve problems related with the arguments of the course. The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration is usually of 1.5 Hours). In place of a single written exam given during the examination sessions, the student may choose instead to substitute the part of exam of the module on Graph Theory and Discrete Mathematics by solving a set of homeworks and small projects, that will be assigned by the teachers during the course. The homeworks are reserved to the students who follow the course in real time. The students who get a positive evaluation in the homeworks will have to pass a reduced written test only on the module of Optimization. The final grade will be the mean of the grades of the two modules.
The outcomes of the exams will be available in the SIFA service through the UNIMIA portal and on the web site of the course.
The outcomes of the exams will be available in the SIFA service through the UNIMIA portal and on the web site of the course
The outcomes of the exams will be available in the SIFA service through the UNIMIA portal and on the web site of the course.
The outcomes of the exams will be available in the SIFA service through the UNIMIA portal and on the web site of the course
Module Graph Theory and Discrete Mathematics
Course syllabus
Recall of Vectors, Matrices, Vector spaces, Vector equations, Linear systems, Linear models, Linear Independence, Linear Transformations, Matrix algebra, Determinants, Inner Product, norm, Orthogonality, Eigenvectors and Eigenvalues.
Singular Value Decomposition. Dimensionality reduction: Principal Components Analysis.
Graph theory: definitions and fundamental concepts. Directed and undirected graphs.
Probability, events and Combinatorics. Finite and discrete time Markov chains. Random walks. Graph theory: fundamental algorithms. Mining social network graphs
Singular Value Decomposition. Dimensionality reduction: Principal Components Analysis.
Graph theory: definitions and fundamental concepts. Directed and undirected graphs.
Probability, events and Combinatorics. Finite and discrete time Markov chains. Random walks. Graph theory: fundamental algorithms. Mining social network graphs
Teaching methods
Face-to-face lectures, tutorials
Teaching Resources
David C. Lay, Steven R. Lay and Judi J. McDonald, Linear Algebra and Its Applications, Pearson, 2016
K. Ruohonen, Graph Theory, Course notes. http://math.tut.fi/~ruohonen/GT_English.pdf (Chapters 1,2,3)
O. Haggstrom, Finite Markov Chains and Algorithmic Applications, London Mathematical Society, 2003. (Chapters1,2,3). http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.9739&rep=rep1&type=pdf
J. Leskovec, A. Rajaraman, J.Ullman, Mining of Massive Datasets, Cambridge University Press, last edition (available online at http://www.mmds.org/ ) - (Chapters 9-10-11).
Lecture Notes of the teachers on the module web site: https://gtdmo.ariel.ctu.unimi.it/v5/home/Default.aspx
K. Ruohonen, Graph Theory, Course notes. http://math.tut.fi/~ruohonen/GT_English.pdf (Chapters 1,2,3)
O. Haggstrom, Finite Markov Chains and Algorithmic Applications, London Mathematical Society, 2003. (Chapters1,2,3). http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.24.9739&rep=rep1&type=pdf
J. Leskovec, A. Rajaraman, J.Ullman, Mining of Massive Datasets, Cambridge University Press, last edition (available online at http://www.mmds.org/ ) - (Chapters 9-10-11).
Lecture Notes of the teachers on the module web site: https://gtdmo.ariel.ctu.unimi.it/v5/home/Default.aspx
Module Optimization
Course syllabus
1.Introduction to complex decision problems: case studies; formal definitions.
2.Mathematical Programming: Karush-Kuhn-Tucker conditions.
3.Multi-objective Programming: Paretian case; Multi-Attribute Utility Theory; Analytic Hierarchy Process; ELECTRE Methods.
4.Uncertain Programming: decision making under ignorance; decision making under risk; decision theory.
2.Mathematical Programming: Karush-Kuhn-Tucker conditions.
3.Multi-objective Programming: Paretian case; Multi-Attribute Utility Theory; Analytic Hierarchy Process; ELECTRE Methods.
4.Uncertain Programming: decision making under ignorance; decision making under risk; decision theory.
Teaching methods
face to face lectures
Teaching Resources
Lecture notes on the module 2 web site: https://homes.di.unimi.it/cordone/courses/courses.html
Module Graph Theory and Discrete Mathematics
SECS-S/06 - MATHEMATICAL METHODS OF ECONOMICS, FINANCE AND ACTUARIAL SCIENCES - University credits: 6
Lessons: 40 hours
Professor:
Micheletti Alessandra
Module Optimization
MAT/09 - OPERATIONS RESEARCH - University credits: 6
Lessons: 40 hours
Professor:
Cordone Roberto
Professor(s)