Stochastic Optimization Methods

Optimization problems arising in practice involve random parameters. For the computation of robust optimal solutions, i.e., optimal solutions being insensitive with respect to random parameter variations, deterministic substitute problems are needed. Based on the distribution of the random data, and using decision theoretical concepts, optimization problems under stochastic uncertainty are converted into deterministic substitute problems. Due to the occurring probabilities and expectations, approximative solution techniques must be applied. Deterministic and stochastic approximation methods and their analytical properties are provided: Taylor expansion, regression and response surface methods, probability inequalities, First Order Reliability Methods, convex approximation/deterministic descent directions/efficient points, stochastic approximation methods, differentiation of probability and mean value functions. Convergence results of the resulting iterative solution procedures are given.

Table of Contents:
Basic Stochastic Optimization Methods: Decision/Control Under Stochastic Uncertainty.- Deterministic Substitute Problems in Optimal Decision Under Stochastic Uncertainty.- Differentiation Methods: Differentiation Methods for Probability and Risk Functions.- Deterministic Descent Directions: Deterministic Descent Directions and Efficient Points.- Semi-Stochastic Approximation Methods: RSM-Based Stochastic Gradient Procedures.- Stochastic Approximation Methods with Different Error Variances.- Technical Applications: Approximation of the Probability of Failure/Survival in Plastic Structural Analysis and Optimal Plastic Design.

Review :
From the reviews: "The aim of the present book is to provide analytical and numerical tools, together with their mathematical foundations, for the approximate computation of robust optimal decisions/designs as needed in concrete engineering/economic applications. a ] The book is well written and the presentation is rigourous and self-contained." (I.M. Stancu-Minasian, Zentralblatt MATH, Vol. 1059 (10), 2005) "The monograph by K. Marti investigates the stochastic optimization approach and presents the deep results of the authora (TM)s intensive research in this field within the last 25 years. a ] The monograph contains many interesting details, results and explanations in semi-stochastic approximation methods and descent algorithms for stochastic optimization problems. a ] Readers interested in these topics will definitely benefit from the monograph." (Stephan Dempe, OR News, 2006) "The book basically goes through the control problem under stochastic uncertainity, which is drawn from the application of engineering and operational research problems. a ] The most important feature of this book is that it has a collection of solution techniques used in optimization methods. a ] More of these applications on different disciplines such as economics a ] made the book accessible for a wider audience and led to a generally more interesting book." (S. Gazioglu, Journal of the Operational Research Society, Vol. 58 (6), 2007)