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Subject: CONVEX ANALYSIS AND OPTIMIZATION (A.A. 2020/2021)

master degree course in MATHEMATICS

Course year 1
CFU 6
Teaching units Unit Convex analysis and optimization
Advanced Theoretical Studies (lesson)
  • TAF: Compulsory subjects, characteristic of the class SSD: MAT/05 CFU: 6
Teachers: Silvia BONETTINI, Michela ELEUTERI
Exam type oral
Evaluation final vote
Teaching language Italiano
Contents download pdf download

Teachers

Michela ELEUTERI
Silvia BONETTINI

Overview

The course's aim is to provide the basis of convex analysis and of the study of convex optimization problems, with the purpose to apply these concepts within the framework of statistical learning and machine learning, which are areas of special interest and research nowadays.

Admission requirements

Basic courses of Mathematical Analysis. It is not required the course of Functional Analysis (Analisi Superiore)

Course contents

First part: foundations of Convex Analysis

General introduction to minimization problems
Lower semicontinuous functions
Lower semicompact functions and minimization problems
Convex sets and convex functions
Lower semicontinuous and convex functions
Proximal theorem
Separation theorem
Characterization of lower semicontinuous and convex functions by means of conjugate functions
Fenchel's theorem
Properties of conjugate functions
Support functions
Subdifferential: properties and calculus rules
Tangent and normal cone

Second part: convex optimization

Optimality conditions

Third part: applications

Teaching methods

Video-recording lessons (asyncronus)

Assessment methods

Colloquium regarding some topics treated in the course and possibly a seminar concerning related topics

Learning outcomes

1. Knowledge and understanding:
at the end of the course a student will have a deep knowledge of foundations of convex analysis, convex optimization methods and their main applications in the direction of statistical learning and machine learning

2. Applying knowledge and understanding:
a deep knowledge of the basis of convex analysis and convex optimization methods will allow the student to set forth autonomous learning paths and strategies in order to efficiently ???

3. Making judgements:
at the end of the course a student should have improved his/her capability of handling theoretical arguments and corresponding laboratory activities

4. Communicating skills:
at the end of the course a student should be able to describe topics presented in the course with an appropriate technical language and a correct mathematical formalism

5. Learning skills:
studying the subject matter, entirely on textbooks and notes in English, should stimulate independent learning skills and the capability of treating connected topics in further detail

Readings

Dispense del corso (a cura delle Prof. Silvia Bonettini e Michela Eleuteri)

J.-P. Aubin: Optima et Equilibria, Springer.