This book presents an introduction to variational analysis, a field which unifies theories and techniques developed in calculus of variations, optimization, and control, and covers convex analysis, nonsmooth analysis, and set-valued analysis. It focuses on problems with constraints, the analysis of which involves set-valued mappings and functions that are not differentiable. Applications of variational analysis are interdisciplinary, ranging from financial planning to steering a flying object. The book is addressed to graduate students, researchers, and practitioners in mathematical sciences, engineering, economics, and finance. A typical reader of the book should be familiar with multivariable calculus and linear algebra. Some basic knowledge in optimization, control, and elementary functional analysis is desirable, but all necessary background material is included in the book.
Author(s): Asen L. Dontchev
Series: Applied Mathematical Sciences 205
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 219
City: Cham, Switzerland
Tags: constrained optimization, set-valued analysis, non smooth analysis, metric regularity, strong regularity, sensitivity, Lipschitz stability, mathematical programming, optimal control
Preface
Contents
0 Notation, Terminology, and Some Functional Analysis
1 Basics in Optimization
2 Continuity of Set-Valued Mappings
3 Lipschitz Continuity of Polyhedral Mappings
4 Metric Regularity
5 Lyusternik–Graves Theorem
6 Mappings with Convex Graphs
7 Derivative Criteria for Metric Regularity
8 Strong Regularity
9 Variational Inequalities over Polyhedral Sets
10 Nonsmooth Inverse Function Theorems
11 Lipschitz Stability in Optimization
12 Strong Subregularity
13 Continuous Selections
14 Radius of Regularity
15 Newton Method for Generalized Equations
16 The Constrained Linear-Quadratic Optimal Control Problem
17 Regularity in Nonlinear Control
18 Discrete Approximations
19 Optimal Feedback Control
20 Model Predictive Control
Bibliographical Remarks and Further Reading
References
List of Symbols
Index
