Design of Distributed and Robust Optimization Algorithms
Simon Michalowsky
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Simon Michalowsky, Design of Distributed and Robust Optimization Algorithms (2020), Logos Verlag, Berlin, ISBN: 9783832587055
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Beschreibung / Abstract
Optimization algorithms are the backbone of many modern technologies. In this thesis, we address the analysis and design of optimization algorithms from a systems theoretic viewpoint. By properly recasting the algorithm design as a controller synthesis problem, we derive methods that enable a systematic design of tailored optimization algorithms.
We consider two specific classes of optimization algorithms: (i) distributed, and (ii) robust optimization algorithms. Concerning (i), we utilize ideas from geometric control in an innovative fashion to derive a novel methodology that enables the design of distributed optimization algorithms under minimal assumptions on the graph topology and the structure of the optimization problem. Concerning (ii), we employ robust control techniques to establish a framework for the analysis of existing algorithms as well as the design of novel robust optimization algorithms with specified guarantees.
We consider two specific classes of optimization algorithms: (i) distributed, and (ii) robust optimization algorithms. Concerning (i), we utilize ideas from geometric control in an innovative fashion to derive a novel methodology that enables the design of distributed optimization algorithms under minimal assumptions on the graph topology and the structure of the optimization problem. Concerning (ii), we employ robust control techniques to establish a framework for the analysis of existing algorithms as well as the design of novel robust optimization algorithms with specified guarantees.
Inhaltsverzeichnis
- BEGINN
- 1 Introduction
- 1.1 Motivation and Background
- 1.2 Problem Formulation
- 1.3 Contributions and Outline
- 2 Design of Distributed Optimization Algorithms
- 2.1 Problem Formulation
- 2.2 Saddle-Point Dynamics
- 2.3 Admissible Lie Bracket Representations
- 2.4 Construction of Distributed Algorithms
- 2.5 Discussions
- 2.6 Summary and Conclusion
- 3 Design of Robust & Structure Exploiting Optimization Algorithms
- 3.1 Problem Formulation
- 3.2 Reformulation in the Robust Control Framework
- 3.3 Robust Optimization Algorithms
- 3.4 Structure Exploiting Algorithms
- 3.5 Discussions
- 3.6 Summary and Conclusion
- 4 Summary and Conclusion
- A Notation and Technical Background
- A.1 Basic Notation
- A.2 Basics of Graph Theory
- A.3 Lie Brackets and Formal Brackets
- A.4 Sequence Spaces and (Linear) Operators
- A.5 Convex Functions and Memoryless Monotone Operators
- A.6 Some Facts Involving Linear Matrix Inequalities (LMIs)
- B Proofs
- B.1 Proofs of Chapter 2
- B.2 Proofs of Chapter 3
- C Additional Material
- C.1 Additional Material to Chapter 2
- C.2 Additional Material to Chapter 3
- List of Symbols
- Bibliography