Convex analysis and optimization in Hadamard spaces 1st Edition by Miroslav Bacak ISBN 978-3110361032 3110361035

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Product details: 

ISBN 10:  3110361035

ISBN 13: 978-3110361032

Author: Miroslav Bacak 

In the past two decades, convex analysis and optimization have been developed in Hadamard spaces. This book represents a first attempt to give a systematic account on the subject.

Hadamard spaces are complete geodesic spaces of nonpositive curvature. They include Hilbert spaces, Hadamard manifolds, Euclidean buildings and many other important spaces. While the role of Hadamard spaces in geometry and geometric group theory has been studied for a long time, first analytical results appeared as late as in the 1990s. Remarkably, it turns out that Hadamard spaces are appropriate for the theory of convex sets and convex functions outside of linear spaces. Since convexity underpins a large number of results in the geometry of Hadamard spaces, we believe that its systematic study is of substantial interest. Optimization methods then address various computational issues and provide us with approximation algorithms which may be useful in sciences and engineering. We present a detailed description of such an application to computational phylogenetics.

The book is primarily aimed at both graduate students and researchers in analysis and optimization, but it is accessible to advanced undergraduate students as well.

Table of contents: 

1. Geometry of Nonpositive Curvature

Geodesic Metric Spaces

Meet Hadamard Spaces

Equivalent Conditions for CAT(0)

2. Convex Sets and Convex Functions

Convex Sets

Convex Functions

Convexity and Probability Measures

3. Weak Convergence in Hadamard Spaces

Existence of Weak Limits

Weak Convergence and Convexity

An Application in Fixed Point Theory

4. Nonexpansive Mappings

Kirszbraun-Valentine Extension

Resolvent of a Nonexpansive Mapping

Strongly Continuous Semigroup

5. Gradient Flow of a Convex Functional

Gradient Flow Semigroup

Mosco Convergence and Its Consequences

Lie-Trotter-Kato Formula

6. Convex Optimization Algorithms

Convex Feasibility Problems

Fixed Point Approximations

Proximal Point Algorithm

7. Probabilistic Tools in Hadamard Spaces

Random Variables and Expectations

Law of Large Numbers

Conditional Expectations

8. Tree Space and Its Applications

Construction of the BHV Tree Space

Owen-Provan Algorithm

Medians and Means of Trees

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Tags: Miroslav Bacak, Convex analysis, optimization in Hadamard