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.

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.

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)

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.

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

Modalità di esame: Prova orale obbligatoria; Elaborato scritto prodotto in gruppo;

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.

Modalità di esame: Prova orale obbligatoria; Elaborato scritto prodotto in gruppo;

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.