Convex Analysis
A.Y. 2018/2019
Learning objectives
Convex Analysis is devoted to study convex sets, convex functions and related extremal problems (minimization, maximization) in finite and infinite dimensional spaces.
The course will remain mainly in the framework of normed spaces.
I shall treat some of the topics listed below (the choice will depend on knowledge and interests of the students), following one of the two lines: "Functional Analysis line" or "Functions and Applications line" (or possibly a "mix" of the two).
The course will remain mainly in the framework of normed spaces.
I shall treat some of the topics listed below (the choice will depend on knowledge and interests of the students), following one of the two lines: "Functional Analysis line" or "Functions and Applications line" (or possibly a "mix" of the two).
Expected learning outcomes
Knowledge of the topics of the course and their application to simple theoretical problems.
Lesson period: Second semester
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
Second semester
Course syllabus
· Convexity in vector spaces: algebraic separation theorems.
· Short account of topological vector spaces: weak topologies.
· Convexity in normed and Banach spaces: topological properties of convex sets, convex hulls, continuity of convex functions, topoloigical speration theorems, extendability of convex Lipschitz functions.
· Finite dimensional convexity: relative interior, Carathéodory theorem and Helly theorem, Jensens's theorem and Hermite-Hadamard theorem, extreme ponts of finite dimensional convex sets (Minkowski thm.).
· Minimization of convex functions.
Functional Analysis line
o Theorems by Krein-Milman and Milman on representability of convex sets by their extreme points, Bauer' maximum principle.
o Boundaries and support points: Bishop-Phelps theorem, theorems of Rodé and James (separable case).
o Duality of convex sets: anihilators and polars.
o Convex series, CS-closed and CS-compact sets.
o Subdifferential of convex functions and differentiability.
o Differentiability of convex functions outside small sets, short account of Asplund spaces.
o Selections of multivalued mapping with convex values, Michael's theorem.
Functions and Applications line
Subdifferential and subdifferential calculus.
Short account on subdifferential and differentiability.
Ekeland's variational principle and It's applications, Bronsted-Rockafellar theorem.
Fenchel duality.
Aproximation of convex functions through infimal convolutions.
Minimax theorems.
· Short account of topological vector spaces: weak topologies.
· Convexity in normed and Banach spaces: topological properties of convex sets, convex hulls, continuity of convex functions, topoloigical speration theorems, extendability of convex Lipschitz functions.
· Finite dimensional convexity: relative interior, Carathéodory theorem and Helly theorem, Jensens's theorem and Hermite-Hadamard theorem, extreme ponts of finite dimensional convex sets (Minkowski thm.).
· Minimization of convex functions.
Functional Analysis line
o Theorems by Krein-Milman and Milman on representability of convex sets by their extreme points, Bauer' maximum principle.
o Boundaries and support points: Bishop-Phelps theorem, theorems of Rodé and James (separable case).
o Duality of convex sets: anihilators and polars.
o Convex series, CS-closed and CS-compact sets.
o Subdifferential of convex functions and differentiability.
o Differentiability of convex functions outside small sets, short account of Asplund spaces.
o Selections of multivalued mapping with convex values, Michael's theorem.
Functions and Applications line
Subdifferential and subdifferential calculus.
Short account on subdifferential and differentiability.
Ekeland's variational principle and It's applications, Bronsted-Rockafellar theorem.
Fenchel duality.
Aproximation of convex functions through infimal convolutions.
Minimax theorems.
Professor(s)