Course info

Курсът е със самозаписване: ключ ВАМИ.

The aim of this course is to present an introduction to variational analysis to be used in a one-semester course for students who are interested to learn and apply modern tools for optimization and control that are usually not covered in textbooks. By ``variational analysis" we mean a mathematical framework that has been developed in the last five decades, which treats in a unified way problems involving set-valued mappings and nondifferentiable functions where classical analysis cannot be applied. It is a good example of mathematics with deep roots, whose rapid growth have been motivated by very practical problems such as financial planning or steering a flying object. The present lecture notes could be useful for students in mathematics, computer science, engineering, operations research and beyond, that would like to have a quick jumpstart in this new kind of analysis.
The material presented should be accessible for students that are familiar with basic calculus and linear algebra and are willing to gain some understanding in applications
of functional analysis.

The material is structured in theorem--proof format but there are also discussions, exercises and applications. The Introduction presents in a condensed form notations and terminology, along with basic facts from analysis. Lecture 1 introduces to optimization problems and shows
necessary optimality condition in the format of generalized equations. In
Lecture 2 the linear-quadratic optimal control problem with control constraints is considered, for which a necessary optimality condition in the form of Pontryagin's principle is shown.

Lecture 3 introduces concepts of continuity of set-valued mappings. As an application, conditions are presented for continuity of the optimal value and optimal solution mappings of a general optimization problem depending on a parameter. Lecture 4 deals with Lipschitz continuity of set-valued mappings, with a special emphasis on polyhedral mappings with application to linear programming, as well as outer Lipschitz continuity of piecewise polyhedral mappings appearing in nonlinear optimization. The next four lectures are devoted to regularity properties of set-valued mappings. Metric regularity of mapping acting in metric spaces, together with the related Aubin property and linear openness, is introduced in Lecture 5.
Two basic result in variational analysis are presented in Lectures 6 and 7, respectively: the Lyusternik-Graves theorem and the Robinson-Ursescu theorem. Strong regularity is introduced in Lecture 8 and applied to a nonlinear programming problem. The next Lecture 9 is devoted to another regularity property, the so-called strong subregularity. The last Lecture 10 broaden the spectрum of applications of the theory presented; it shows quadratic convergence of Newton's method applied to generalized equations under the regularity conditions discussed in the preceding lectures.

In this course we do not cover generalized differentiation, which is sometimes considered as
a part of variational analysis. Introducing technical constructions of derivatives of nondifferentiable functions and set-valued mappings, even
on a basic level, would go beyond the scope of the course.

Some parts of these notes contain appropriately adapted
material from the book [A. L. Dontchev, R. T. Rockafellar, Implicit functions and solution mappings, 2nd edition, Springer 2014] but the notes are by no means a substitute of this book. Actually, that book would be a good next step to gain a deeper understanding of the area.

The lecture notes attached to the course are subject to changes, additions, adjustments, hopefully improvements, and the author will be grateful for any comments, opinions and suggestions; please email feedback to doburden@gmail.com and dontchev@umich.edu.

The specific presentation of the lectures on-line is to be determined at a later stage
after interactions with prospective students.