Optimization is a technology that can be used to devise effective decisions or predictions in a variety of contexts, ranging from production planning, to engineering design, finance, machine learning and data science, to mention just a few. In simplified terms, the process for reaching a decision starts with a phase of construction of a suitable mathematical model for a concrete problem, followed by a phase where the model is solved by means of suitable numerical algorithms. An optimization model typically requires the specification of a quantitative objective criterion of goodness for our decision, which we wish to maximize (or, alternatively, a criterion of cost, which we wish to minimize), as well as the specification of constraints, representing the physical limits of our decision actions, budgets on resources, design requirements than need be met, etc. An optimal design is one who gives the best possible objective value, while satisfying all problem constraints. While a generic optimization approach typically leads to computational problems that are very hard to solve in practice, convex problems, which are the core of this course, possess special properties that make them amenable to very efficient numerical solution techniques. This course therefore concentrates on recognizing, modeling and solving convex optimization problems that arise in several scientific contexts (engineering, finance, control, data science, etc.), and in showing the relevance of these methodologies in practical applications.

To give students the tools and training to recognize convex optimization problems that arise in engineering and other branches of science; to present the basic theory of such problems, concentrating on results that are useful in computation; to give students an understanding of how such problems are solved, and some experience in solving them.

Good knowledge of linear algebra, geometry, analysis and some exposure to probability. Exposure to numerical computing, optimization, systems and control theory, and application fields is helpful but not strictly required.

Introduction, convex sets and convex functions. Optimization problems in standard form, optimality criteria. Systems of linear equations, Least Squares (LS), Linear Programming (LP), Ell-one norm optimization, Chebychev approximation. Application examples: generation of force/torque via thrusters, uniform illumination of patch surfaces, etc. Quadratic Programming (QP) and Second Order Cone Programming (SOCP). Application examples (e.g. FIR filter design, antenna array design, sidelobe level minimization, etc.). Linear Matrix Inequalities (LMI) and semidefinite programming (SDP). Geometric programming (GP). Software tools (CVX and/or YALMIP). Applications: data-fitting, approximation and estimation, uncertain and robust Least Squares, portfolio optimization. Geometrical problems: containment of poyhedra, classification, Lowner-John ellipsoids, linear discrimination, support vector machines. Focus topics (mutually exclusive, to be offered alternatively over years): - LMIs in systems and control theory. - Sparse optimization and compressed sensing. - Convex optimization in Finance. - Convex optimization in algebraic geometry, global polynominal positivity (positivstellensaatz). - Network optimization, distributed optimization.

The course is organized in a series of lectures (about 1/3 of the course) and computer lab exercises and practice sessions (about 2/3 of the course).

G.C. Calafiore and L. El Ghaoui, Optimization Models, Cambridge Univ. Press, 2014. S. Boyd and L. Vandenberghe; Convex Optimization, Cambridge Univ. Press, 2004.

Modalità di esame: Prova scritta (in aula); Progetto di gruppo;

Exam: Written test; Group project;

... The final exam consists in a written test, which will contain a mixture of methodological questions and numerical exercises (to be executed with pen and paper, use of a calculator is allowed) and which will take the form of multiple-choice questionar. For students who fully participate in the lab sessions an optional alternate form of final exam is offered, which consists of a written report on an applicative project assigned by the instructor. In this case, the students can work on the project in groups of at most five persons; a reasonable time will be allowed for completing the report (typically, one week), and each group is requested to give a public presentation of their results in front of the class. The evaluation and corresponding score will be based on the results of lab assignments and factors including technical correctness of the report, organization of the presentation, and autonomy in the report development. No oral exams are foreseen, besides the oral presentation of the report.

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.