(in Polish) Analiza wypukła 1000-1M21AWP
We will start with the characterization of convex functions and sets and define the basic operations on these objects. Next, we will discuss their topological properties. After that, we will focus on the duality between points and hypersurfaces, and introduce convex conjugation (Fenchel transform) and polarity operations. When examining convex cones, we will also focus for a moment on the relationship with the theory of norms. We will further prove The Carathéodory representation theorem for points in the convex hull of some set and also deal with the concept of extreme points. We shall then move on to the differentiability of convex functions, define the subgradient, discuss the Legendre transform, and examine its relationship to convex conjugation. We will examine when the gradient of a convex function defines the homeomorphism of its domain with the domain of the conjugate. Perhaps it will also be possible to prove The Alexandrov theorem about the existence of a second Taylor polynomial for a convex function in almost every point of its domain. Finally, if time permits, we will tackle the topic of convex minimization with convex constraints and the min-max problem for the convex-concave function. In particular, we will prove the Fenchel duality theorem.
In the exercises, apart from solving the tasks illustrating and supplementing the lecture, we will look at a certain generalization of convex sets, namely sets of positive reach.
Main fields of studies for MISMaP
Type of course
Mode
Prerequisites
Prerequisites (description)
Learning outcomes
* Knowledge of basic tools and theorems for finite dimensional convex analysis.
* Awareness of the relationship between convex analysis and calculus of variations and optimization problems.
* Understanding the relationship between convex analysis, convex geometry, and the theory of normed spaces.
* Knowledge of some applications, e.g. for studying sets of positive reach.
Assessment criteria
To be admitted to the exam one needs to present, in exercise sessions, certain number of solutions of given problems (homework).
The final exam shall be oral but the list of questions / topics / problems will be given in advance (at the end of the semester).
Bibliography
Primary:
* "Convex analysis" R. T. Rockafellar
* "Fundatnentals of Convex Analysis" J-B. Hiriart-Urruty, C. Lemarechal
Secondary:
* "Minkowski Geometry" A. C. Thompson
* "Lectures on Convex Geometry" D. Hug, W. Weil
* "Variational Analysis" R. T. Rockafellar, R. J-B. Wets
* "Convex Functions and their Applications: A Contemporary Approach" C. P. Niculescu, L-E. Persson
* "Curvature measures" H. Federer
* "User’s guide to viscosity solutions of second order partial differential equations" M. Crandall, H. Ishii, P. Lions
* "Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets" A.D. Rosa, S. Kolasiński, M. Santilli
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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