Stochastic Optimization Methods

Transcription

1 Stochastic Optimization Methods

2 Kurt Marti Stochastic Optimization Methods With 14 Figures 4y Springer

3 Univ. Professor Dr. sc. math. Kurt Marti Federal Armed Forces University Munich Aero-Space Engineering and Technology Neubiberg/Munich Germany Cataloging-in-Publication Data Library of Congress Control Number: ISBN Springer Berlin Heidelberg New York 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-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com Springer Berlin Heidelberg 2005 Printed in Germany 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. Hardcover-Design: Erich Kirchner, Heidelberg SPIN / Printed on acid-free paper

4 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, 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, 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. In order to cope with these uncertainties, a basic procedure in the engineering/economic practice is to replace first the unknown parameters by some chosen nominal values, e.g. estimates, 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 omitted to a large extent by taking into account already at the planning stage the possible consequences of the tracking errors and the known prior and sample information about the random data of the problem. Hence, instead of relying on ordinary deterministic parameter optimization methods - based on some nominal parameter values - and applying then just some correction actions, stochastic optimization methods should be applied: Incorporating the stochastic parameter variations into the optimization process, expensive and increasing online correction expenses can be omitted or at least reduced to a large extent.

5 VI Preface Consequently, for the computation of robust optimal decisions/designs, i.e., optimal decisions which are insensitive with respect to random parameter variations, appropriate deterministic substitute problems must be formulated first. Based on decision theoretical principles, these substitute problems depend on probabilities of failure/success and/or on more general expected cost/loss terms. Since probabilities and expectations are defined by multiple integrals in general, the resulting often nonlinear and also nonconvex deterministic substitute problems can be solved by approximative methods only. Hence, the analytical properties of the substitute problems are examined, and appropriate deterministic and stochastic solution procedures are presented. The aim of the present book is therefore to provide analytical and numerical tools - including their mathematical foundations - for the approximate computation of robust optimal decisions/designs as needed in concrete engineering/economic applications. Two basic types of deterministic substitute problems occur mostly in practice: * Reliability-Based Optimization Problems: Minimization of the expected primary costs subject to expected recourse cost constraints (reliability constraints) and remaining deterministic constraints, e.g. box constraints. In case of piecewise constant cost functions, probabilistic objective functions and/or probabilistic constraints occur; * Expected Total Cost Minimization Problems: Minimization of the expected total costs (costs of construction, design, recourse costs, etc.) subject to the remaining deterministic constraints. Basic methods and tools for the construction of appropriate deterministic substitute problems of different complexity and for various practical applications are presented in Chapter 1 and part of Chapter 2 together with fundamental analytical properties of reliability-based optimization/expected cost minimization problems. Here, a main problem is the analytical treatment and numerical computation of the occurring expectations and probabilities depending on certain input variables. For this purpose deterministic and stochastic approximation methods are provided which are based on Taylor expansion methods, regression and response surface methods (RSM), probability inequalities, differentiation formulas for probability and expectation value functions. Moreover, first order reliability methods (FORM), Laplace expansions of integrals, convex approximation/deterministic descent directions/efficient points, (hybrid) stochastic gradient methods, stochastic approximation procedures are considered. The mathematical properties of the approximative problems and iterative solution procedures are described. After the presentation of the basic approximation techniques in Chapter 2, in Chapter 3 derivatives of probability and mean value functions are obtained by the following methods: Transformation method, stochastic completion and transformation, orthogonal function series expansion. Chapter 4

6 Preface shows how nonconvex deterministic substitute problems can be approximated by convex mean value minimization problems. Depending on the type of loss functions and the parameter distribution, feasible descent directions can be constructed at non-efficient points. Here, efficient points, determined often by linear/quadratic conditions involving first and second order moments of the parameters, are candidates for optimal solutions to be obtained which much larger effort. The numerical/iterative solution techniques presented in Chapter 5 are based on hybrid stochastic approximation, hence, methods using not only simple stochastic (sub) gradients, but also deterministic descent directions and/or more exact gradient estimators. More exact gradient estimators based on Response Surface Methods (RSM) are described in the first part of Chapter 5. The convergence in the mean square sense is considered, and results on the speed up of the convergence by using improved step directions at certain iteration points are given. Various extensions of hybrid stochastic approximation methods are treated in Chapter 6 on stochastic approximation methods with changing variance of the estimation error. Here, second order search directions, based e.g. on the approximation of the Hessian, and asymptotic optimal scalar and matrix valued step sizes are constructed. Moreover, using different types of probabilistic convergence concepts, related convergence properties and convergence rates are given. Mathematical definitions and theorems needed here are given in the Appendix. Applications to the approximate computation of survival/failure probabilities for elastoplastic mechanical structures and its application to stochastic structural optimization are given in the last Chapter 7. The reader of this book needs some basic knowledge in linear algebra, multivariate analysis and stochastics. Realizing this monograph, the author was supported by several collaborators: First of all I would like to thank Ms Elisabeth Lossl from the Institute for Mathematics and Computer Sciences for the excellent I^TEXtypesetting of the whole manuscript. Without her very precise and careful work, this project could not have been realized. I also owe thanks to my former collaborators, Dr. Andreas Aurnhammer, for further I^TEXsupport and for providing English translation of some German parts of the original manuscript, and to Thomas Platzer for proof reading of some parts of the book. Last but not least I am indebted to Springer-Verlag for inclusion the book into the Springer-program. I would like to thanks especially to the Publishing Director Economics and Management Science of Springer-Verlag, Dr. Werner A. Miiller, for his really very long patience until the completion of this book. Munich, August 2004 Kurt Marti VII

7 Contents Part I Basic Stochastic Optimization Methods 1 Decision/Control Under Stochastic Uncertainty Introduction Deterministic Substitute Problems: Basic Formulation Minimum or Bounded Expected Costs Minimum or Bounded Maximum Costs (Worst Case) Deterministic Substitute Problems in Optimal Decision Under Stochastic Uncertainty Optimum Design Problems with Random Parameters Deterministic Substitute Problems in Optimal Design Deterministic Substitute Problems in Quality Engineering Basic Properties of Substitute Problems Approximations of Deterministic Substitute Problems in Optimal Design Approximation of the Loss Function Regression Techniques, RSM Taylor Expansion Methods Applications to Problems in Quality Engineering Approximation of Probabilities - Probability Inequalities Bonferroni-Type Inequalities Tschebyscheff-Type Inequalities FORM (First Order Reliability Methods) Construction of State Functions in Structural Analysis and Design Plastic Analysis and Optimal Plastic Design Optimal Elastic Design 41

8 X Contents Part II Differentiation Methods Differentiation Methods for Probability and Risk Functions Introduction Transformation Method: Differentiation by Using an Integral Transformation Representation of the Derivatives by Surface Integrals The Differentiation of Structural Reliabilities Extensions More General Response (State) Functions Computation of Probabilities and its Derivatives by Asymptotic Expansions of Integral of Laplace Type Computation of Structural Reliabilities and its Sensitivities Numerical Computation of Derivatives of the Probability Functions Arising in Chance Constrained Programming Integral Representations of the Probability Function P(x) and its Derivatives Orthogonal Function Series Expansions I: Expansions in Hermite Functions, Case m = Integrals over the Basis Functions and the Coefficients of the Orthogonal Series Estimation/Approximation of P(x) and its Derivatives The Integrated Square Error (ISE) of Deterministic Approximations Orthogonal Function Series Expansions II: Expansions in Hermite Functions, Case m > Orthogonal Function Series Expansions III: Expansions in Trigonometric, Legendre and Laguerre Series Expansions in Trigonometric and Legendre Series Expansions in Laguerre Series 93 Part III Deterministic Descent Directions 4 Deterministic Descent Directions and Efficient Points Convex Approximation Approximative Convex Optimization Problem Computation of Descent Directions in Case of Normal Distributions Descent Directions of Convex Programs Solution of the Auxiliary Programs Efficient Solutions (Points) 115

9 Contents Necessary Optimality Conditions Without Gradients Existence of Feasible Descent Directions in Non-Efficient Solutions of (4.9a,b) Descent Directions in Case of Elliptically Contoured Distributions Construction of Descent Directions by Using Quadratic Approximations of the Loss Function 122 XI Part IV Semi-Stochastic Approximation Methods 5 RSM-Based Stochastic Gradient Procedures Introduction Gradient Estimation Using the Response Surface Methodology (RSM) The Two Phases of RSM The Mean Square Error of the Gradient Estimator Estimation of the Mean Square (Mean Functional) Error The Argument Case The Criterial Case Convergence Behavior of Hybrid Stochastic Approximation Methods Asymptotically Correct Response Surface Model Biased Response Surface Model Convergence Rates of Hybrid Stochastic Approximation Procedures Fixed Rate of Stochastic and Deterministic Steps Lower Bounds for the Mean Square Error Decreasing Rate of Stochastic Steps Stochastic Approximation Methods with Changing Error Variances Introduction Solution of Optimality Conditions General Assumptions and Notations Interpretation of the Assumptions Notations and Abbreviations in this Chapter Preliminary Results Estimation of the Quadratic Error Consideration of the Weighted Error Sequence Further Preliminary Results General Convergence Results Convergence with Probability One Convergence in the Mean Convergence in Distribution 195

10 XII Contents 6.6 Realisation of Search Directions Y n Estimation of G* Update of the Jacobian Estimation of Error Variances Realization of Adaptive Step Sizes Optimal Matrix Step Sizes Adaptive Scalar Step Size A Special Class of Adaptive Scalar Step Sizes Convergence Properties Examples for the Function Q n (r) Optimal Sequence (w n ) Sequence (K n ) 247 Part V Technical Applications 7 Approximation of the Probability of Failure/Survival in Plastic Structural Analysis and Optimal Plastic Design Introduction Probability of Survival/Failure p s,pf Approximation of p a,pf by Linearization of the Transformed Limit State Function The Origin Lies in the Transformed Safe Domain The Origin Lies in the Transformed Failure Domain Computation of the /?-Point z* Trusses Special Case Reliability-Based Design Optimization (RBDO) Necessary Optimality Conditions for the /?-Point Duality Relations 270 Part VI Appendix A Sequences, Series and Products 275 A.I Mean Value Theorems for Deterministic Sequences 275 A.2 Iterative Solution of a Lyapunov Matrix Equation 283 B Convergence Theorems for Stochastic Sequences 287 B.I A Convergence Result of Robbins-Siegmund 287 B.I.I Consequences 287 B.2 Convergence in the Mean 290 B.3 The Strong Law of Large Numbers for Dependent Matrix Sequences 292 B.4 A Central Limit Theorem for Dependent Vector Sequences

11 Contents XIII C Tools from Matrix Calculus 295 C.I Miscellaneous 295 C.2 The v. Mises-Procedure in Case of Errors 296 References 301 Index 309

12 Part I Basic Stochastic Optimization Methods