CONVEX ANALYSIS

Course objectives

Knowledge and understanding Aim of the course is to provide basic notions and skills of Convex Analysis in finite dimensional spaces. Particular attention will be devoted to the analytical aspects of convexity and their use in geometric and optimization problems. Application skills To be able to deal with problems involving convexity, in particular: characterizations of convexity, convex inequalities, regularity and monotony properties of convex functions, subdifferentials, separation of convex sets, convex optimization. Autonomy of judgment To have essential tools for successive approach to functional analysis, partial differential equations, control theory and mathematical programming. To be able to autonomously solve new problems, applying mathematical tools to phenomena or processes to be encountered in University studies or subsequent working activities. Communication skills To know how to communicate using properly mathematical language. Ability to learn To be able to deepen autonomously some arguments introduced during the course.

Channel 1
GIULIO GALISE Lecturers' profile

Program - Frequency - Exams

Course program
Convex functions of one variable: almost everywhere differentiability, local Lipschitz continuity, supporting lines, maximum principle and convex inequalities (e.g. Jensen, aritmetic/geometric/harmonic mean) Convex functions of several variables: basic properties, distance function, local Lipschitz continuity, subdifferential, supporting hyperplanes, directional derivatives and critical points. Hahn-Banach theorem (analytic form) in finite dimension and its consequences, in particular the characterization of convex functions in terms of non-empty subdifferential. separations of convex sets and applications. Differentiablility almost everywhere of convex functions . Geometry of convex sets: basic (algebraic, topological ) properties of convex sets, relevant examples ( polyhedron, simplex, convex envelope), Carathéodory's theorem, Hahn-Banach's theorem (geometric forms) and separation of convex sets, Brunn-Minowski and isoperimetric inequalities POSSIBLE FURTHER INSIGHTS (if time permits) Geometry of convex sets, further results: e.g. Helly and Krein-Milman theorems Optimization of concave\convex functions, constrained extremum problems, Kuhn-Tucker conditions Conjugate Convex Functions and Fenchel- Legendre duality
Prerequisites
The course requires familiarity with differential and integral calculus for real functions of one and several real variables and with main topics of Linear Algebra. No preparatory course is mandatory.
Books
Lecture notes Further bibliographic references: 1.Borwein-Wanderwer, Convex Functions (Constructions, Characterizations and Counterexamples), Cambridge 2.Krantz, Convex Analysis, CRC Press 3.Mordukhovich-Nam, An Easy Path to Convex Analysis and Applications, M&C Publisher 4.Rockafellar, Convex Analysis, Princeton University Press
Frequency
Attendance is not mandatory , but strongly encouraged.
Exam mode
The exam is divided into a written tes (with numerical and theoretical exercises) and an oral exam.
Bibliography
Borwein-Wanderwer, Convex Functions (constructions, characterizations and counterexamples), Cambridge I. Capuzzo Dolcetta, F. Lanzara, A. Siconolfi, Lezioni di Ottimizzazione. Edizioni Nuova Cultura (2013) Dacorogna, Introduction to the Calculus of Variations, Springer Krantz, Convex analysis, CRC Press Mordukhovich-Nam, An easy path to convex analysis and applications, M&C Publisher Rockafellar, Convex Analysis, Princeton Univeristy Press
Lesson mode
Lectures and exercises sessions
  • Lesson code10611767
  • Academic year2025/2026
  • CourseMathematics
  • CurriculumGenerale
  • Year3rd year
  • Semester2nd semester
  • SSDMAT/05
  • CFU6