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Numerical optimization for large scale problems and Stochastic Optimization
02TXDSM, 02TXDNG
A.A. 2024/25
Course Language
Inglese
Degree programme(s)
Master of science-level of the Bologna process in Data Science And Engineering - Torino
Master of science-level of the Bologna process in Ingegneria Matematica - Torino
Course structure
| Teaching | Hours |
|---|---|
| Lezioni | 60 |
| Esercitazioni in laboratorio | 20 |
Lecturers
| Teacher | Status | SSD | h.Les | h.Ex | h.Lab | h.Tut | Years teaching |
|---|---|---|---|---|---|---|---|
| Pieraccini Sandra | Professore Ordinario | MATH-05/A | 40 | 0 | 0 | 0 | 6 |
Co-lectures
Context
| SSD | CFU | Activities | Area context | MAT/08 MAT/09 |
6 2 |
C - Affini o integrative C - Affini o integrative |
Attività formative affini o integrative Attività formative affini o integrative |
|---|
2024/25
The course aims at introducing some of the main tools for supporting machine learning algorithms. In particular, the focus will be on the computation of the numerical solution of very large scale nonlinear optimization problems. Both unconstrained and constrained problems will be tackled, describing different methods suitable for the various problems according to their classification (e.g., nonlinear least square problems, quadratic programming problems, purely nonlinear problems...). Foundations of stochastic dynamic programming will be also introduced.
The course aims to introduce some of the main tools for supporting machine learning algorithms. In particular, the focus will be on the computation of the numerical solution of large-scale nonlinear optimization problems. Both unconstrained and constrained problems will be tackled, describing different methods suitable for the various problems according to their classification (e.g., nonlinear least square problems, quadratic programming problems, purely nonlinear problems...). Foundations of stochastic dynamic programming will also be introduced.
Knowledge of the main numerical methods for solving large scale numerical optimization problems, according to the problem type.
Ability to choose and correctly apply software for different numerical optimization problems, with a critical investigation of the results obtained.
The student is expected to acquire knowledge of the main numerical methods for solving large-scale numerical optimization problems, according to the problem type, to develop the ability to choose and correctly apply software for different numerical optimization problems, and to perform a critical investigation of the results obtained.
Knowledge of basic linear algebra and calculus, as well as a basic coding ability.
Knowledge of basic linear algebra and calculus, as well as a basic coding ability.
Course syllabus
• Convex optimization:
- gradient descent method; conjugate gradient method
- Numerical differentiation
- Newton and quasi-Newton methods
- Globalization techniques
- Alternating direction method of multipliers (ADMM)
• Constrained optimization:
- Interior point methods
- Projected gradient method
- Active set
• Stochastic optimization
- Static simulation-based optimization (parametric optimization)
- Dynamic simulation-based optimization (control optimization)
Unconstrained optimization:
- gradient and conjugate gradient methods for linear systems
- descent methods
- Newton, quasi-Newton, and modified Newton methods
- Gauss-Newton method
- line-search strategies
Constrained optimization:
- Optimality conditions
- Projected gradient method
- Penalty methods
- Alternating direction method of multipliers (ADMM)
Stochastic optimization:
- Static simulation-based optimization (parametric optimization)
- Dynamic simulation-based optimization (control optimization)
Theoretical lectures and practice classes. Theoretical lectures are devoted to the presentation of the topics, with definitions, properties, introductory examples. The practice classes are devoted to train the students’ abilities to solve problems and exercises and to perform computations and simulations with common tools.
The course is organized into theoretical lectures and practice classes. Theoretical lectures are devoted to the presentation of the topics, with definitions, properties, and introductory examples. The practice classes are devoted to training the students’ abilities to solve problems and exercises and to perform computations and simulations with common tools.
Slides presented during lesson will be made avalaible through the Portale della Didattica. Other material will be suggested in class and, if possible, made avalaible through the Portale della Didattica.
Suggested textbook:
J. Nocedal, S. J. Wright, Numerical Optimization, Springer, 2006
A. Gosavi, Simulation-Based Optimization, 2nd edition, Springer, 2015
Slides presented during lesson will be made avalaible through the Portale della Didattica. Other material will be suggested in class and, if possible, made avalaible through the Portale della Didattica.
Suggested textbook:
J. Nocedal, S. J. Wright, Numerical Optimization, Springer, 2006
A. Gosavi, Simulation-Based Optimization, 2nd edition, Springer, 2015
Slides; Libro di testo; Esercizi risolti; Esercitazioni di laboratorio risolte; Video lezioni tratte da anni precedenti;
Lecture slides; Text book; Exercise with solutions ; Lab exercises with solutions; Video lectures (previous years);
E' possibile sostenere l’esame in anticipo rispetto all’acquisizione della frequenza
You can take this exam before attending the course
Modalità di esame: Prova orale obbligatoria; Elaborato scritto prodotto in gruppo;
Exam: Compulsory oral exam; Group essay;
...
The exam is based on the reports on homeworks assigned during the course and on an oral exam.
In detail, three homeworks will be assigned during the course, consisting both in exercises, aimed at evaluating the students in applying the methods presented, and in a practical implementation/application of the methods described during lectures.
An oral test will then consist of two parts:
a) a discussion of the submitted reports, aimed at testing the depth of the students’ understanding of the subjects and their ability to explain, defend, reflect, critically evaluate, and possibly improve their work;
b) a presentation of a topic studied in the course, covering both theoretical aspects and possibly their implementation issues and examples of fields of applications.
Grading: the maximum grade for the homework reports, finalized upon the discussion detailed at point (a) above, is of 14 points. The maximum grade for part (b) of the oral test is 18 points. The final course grade is then obtained by summing up the final grades for the reports and the grade for part (b) of the oral test.
Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.
Exam: Compulsory oral exam; Group essay;
The exam is based on homework reports assigned during the course and on an oral discussion.
In detail, some problems will be assigned during the semester, aimed at testing students' ability to identify the correct strategy to solve the problem, apply the methods presented during the lectures, and comment on the practical behavior of the methods applied on some given test cases. The students have to choose two assignments among the proposed ones and write a report for each assignment. Assignment 1 has to be chosen among the stochastic optimization proposals;
Assignment 2 has to be chosen among the unconstrained/constrained optimization proposals.
The report should be the result of teamwork, with each team made of 2 or 3 students.
The oral discussion will be then organized as follows:
a1) a discussion on Assignment 1, aimed at testing the depth of the student's understanding of the subjects and their ability to explain, defend, reflect, critically evaluate, and possibly improve their work;
a2) a discussion on Assignment 2, aimed at testing the depth of the student's understanding of the subjects and their ability to explain, defend, reflect, critically evaluate, and possibly improve their work;
a3) a discussion on other topics studied in the course, with a special focus on (but not limited to) topics not addressed by the report. Both theoretical aspects and implementation issues and examples of fields of applications will be discussed.
The grading of parts a1) and a2) will be based both on the quality of the reports and on their oral defense. The maximum grading attainable is:
- 8 points for part (a1)
- 12 points for part (a2)
- 12 points for part (a3)
The final course grade is obtained by summing up the grades obtained for each part (a1)-(a3).
In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.