RSM-Based Stochastic Gradient Procedures
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Kurt Marti
Stochastic Optimization Methods Second edition
123
Univ. Prof. Dr. Kurt Marti Federal Armed Forces University Munich Department of Aerospace Engineering and Technology 85577 Neubiberg/München Germany [email protected]
ISBN: 978-3-540-79457-8
e-ISBN: 978-3-540-79458-5
Library of Congress Control Number: 2008925523 c 2008 Springer-Verlag Berlin Heidelberg ° This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 9 8 7 6 5 4 3 2 springer.com
Preface
Optimization problems in practice depend mostly on several model parameters, noise factors, uncontrollable parameters, etc., which are not given fixed quantities at the planning stage. Typical examples from engineering and economics/operations research are: Material parameters (e.g. elasticity moduli, yield stresses, allowable stresses, moment capacities, and specific gravity), external loadings, friction coefficients, moments of inertia, length of links, mass of links, location of center of gravity of links, manufacturing errors, tolerances, noise terms, demand parameters, technological coefficients in input-output functions, cost factors, etc.. Due to several types of stochastic uncertainties (physical uncertainty, economic uncertainty, statistical uncertainty, and model uncertainty) these parameters must be modelled by random variables having a certain probability distribution. In most cases at least certain moments of this distribution are known. To cope with these uncertainties, a basic procedure in engineering/economic practice is to replace first the unknown parameters by some chosen nominal values, e.g. estimates or guesses, of the parameters. Then, the resulting and mostly increasing deviation of the performance (output, behavior) of the structure/system from the prescribed performance (output, behavior), i.e., the “tracking error”, is compensated by (online) input corrections. However, the online correction of a system/structure is often time-consuming and causes mostly increasing expenses (correction or recourse costs). Very large recourse costs may arise in case of damages or failures of the plant. This can be avoided to a large extent by taking into account at the planning stage the poss
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