Optimization Vol. 1 – Optimality conditions, elements of convex analysis and duality
Authors
Description
This volume contains a rigorous and complex treatment of first- and second-order optimality conditions, as well as topics in Convex Analysis and Duality theory. It is based on courses that the authors have been teaching at Moscow University and IMPA. An important feature of the book is that it is self-contained – all claims are proven completely and with complete mathematical rigor, without appealing to external facts or results. The book contains many pictures to facilitate exposition and many exercises to fix knowledge. Some of the exercises are easy to solve; others are non-trivial and can be used as a challenge for readers.
Target audience
Higher education
Name: Optimization Vol. 1 – Optimality conditions, elements of convex analysis and duality
Author(s): Alexey Izmailov e Mikhail Solodov
Pages: 256
Publication: IMPA, 2020
ISBN: 978-65990528-0-4
Edition: 4
1 Introduction to Optimization
1.1 Definitions and some basic facts
1.2 Existence of global solutions
1.3 Optimal conditions for unconstrained problems
1.4 Optimum conditions for constraint problems
2 Problems with Equality Constraints
2.1 Tangent in the case of equality restrictions
2.2 Lagrange’s optimum conditions
2.3 Second-order optimality conditions
3 Elements of Convex Analysis
3.1 Definitions and basic facts of convexity
3.2 Convex Assemblies. Separation theorems
3.2.1 Basic Properties of Convex Sets
3.2.2 The projection operator
3.2.3 Separation theorems
3.2.4 Extreme points
3.3 Alternative theorems
3.4 Convex functions
3.4.1 Basic Properties of Convex Functions
3.4.2 Differentiable convex functions
3.4.3 Non-differentiable convex functions
4 Problems with Restrictions of Equality and Inequality
4.1 Tangent in the case of equality and inequality constraints
4.2 Karush-Kuhn-Tucker optimum conditions
4.3 Second-order optimum conditions
5 Elements of the Theory of Duality
5.1 Duality in linear programming
5.2 Duality for a general problem
Bibliography
