Nonlinear Optimization with Engineering Applications

This book, like its companion volume Nonlinear Optimization with Financial Applications, is an outgrowth of undergraduate and po- graduate courses given at the University of Hertfordshire and the University of Bergamo. It deals with the theory behind numerical methods for nonlinear optimization and their application to a range of problems in science and engineering. The book is intended for ?nal year undergraduate students in mathematics (or other subjects with a high mathematical or computational content) and exercises are provided at the end of most sections. The material should also be useful for postg- duate students and other researchers and practitioners who may be c- cerned with the development or use of optimization algorithms. It is assumed that readers have an understanding of the algebra of matrices and vectors and of the Taylor and mean value theorems in several va- ables. Prior experience of using computational techniques for solving systems of linear equations is also desirable, as is familiarity with the behaviour of iterative algorithms such as Newton’s methodfor nonlinear equations in one variable. Most of the currently popular methods for continuous nonlinear optimization are described and given (at least) an intuitive justi?cation. Relevant convergence results are also outlined and we provide proofs of these when it seems instructive to do so. This theoretical material is complemented by numerical illustrations which give a ?avour of how the methods perform in practice.

Introducing Optimization.- One-variable Optimization.- Applications in n Variables.- n-Variable Unconstrained Optimization.- Direct Search Methods.- Computing Derivatives.- The Steepest Descent Method.- Weak Line Searches and Convergence.- Newton and Newton-like Methods.- Quasi-Newton Methods.- Conjugate Gradient Methods.- ASummary of Unconstrained Methods.- Optimization with Restrictions.- Larger-Scale Problems.- Global Unconstrained Optimization.- Equality Constrained Optimization.- Linear Equality Constraints.- Penalty Function Methods.- Sequential Quadratic Programming.- Inequality Constrained Optimization.- Extending Equality Constraint Methods.- Barrier Function Methods.- Interior Point Methods.- A Summary of Constrained Methods.- The OPTIMA Software.