Optimization of Polynomials in Non-Commuting Variables
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will f
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Sabine Burgdorf Igor Klep Janez Povh
Optimization of Polynomials in Non-Commuting Variables
123
SpringerBriefs in Mathematics
Series Editors Nicola Bellomo Michele Benzi Palle Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel George Yin Ping Zhang
SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030
Sabine Burgdorf • Igor Klep • Janez Povh
Optimization of Polynomials in Non-Commuting Variables
123
Sabine Burgdorf Centrum Wiskunde & Informatica Amsterdam, The Netherlands
Janez Povh Faculty of Information Studies in Novo Mesto Novo Mesto, Slovenia
Igor Klep Department of Mathematics The University of Auckland Auckland, New Zealand
ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-319-33336-6 ISBN 978-3-319-33338-0 (eBook) DOI 10.1007/978-3-319-33338-0 Library of Congress Control Number: 2016938173 Mathematics Subject Classification (2010): 90C22, 14P10, 90C26, 08B20, 13J30, 47A57 © The Author(s) 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland
Introduction
Optimization problems involving polynomial data arise across many sciences, e.g., in control theory [Che10, HG05, Sch06], operations research [Sho90, Nie09], statistics and probability [Las09], combinatorics and graph theory [LS91, AL12], computer science [PM81], and elsewhere. They are however difficult to solve. For example, very simple instances of polynomial optimization
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