TUM Library

Chapter 1. Nonlinear Optimization Overview1.1 Introduction1.2 Modeling1.3 ExercisesChapter 2. Review of Single Variable Calculus Topics 2.1 Limits 2.2 Continuity 2.3 Differentiation 2.4 ConvexityChapter 3. Single Variable Optimization 3.1 Introduction 3.2 Optimization Applications 3.3 Optimization ModelsConstrained Optimization by CalculusChapter 4. Single Variable Search Methods4.1 Introduction 4.2 Unrestricted Search 4.3 Dichotomous Search 4.4 Golden Section Search 4.5 Fibonacci Search 4.6 Newton's Method 4.7 Bisection Derivative SearchChapter 5. Review of MV Calculus Topics5.1 Introduction, Basic Theory, and Partial Derivatives5.2 Directional Derivatives and The GradientChapter 6. MV Optimization 6.1 Introduction 6.2 The Hessian 6.3 Unconstrained OptimizationConvexity and The Hessian MatrixMax and Min Problems with Several VariablesChapter 7. Multi-variable Search Methods 7.1 Introduction7.2 Gradient Search7.3 Modified Newton's MethodChapter 8. Equality Constrained Optimization: Lagrange Multipliers 8.1 Introduction and Theory 8.2 Graphical Interpretation 8.3 Computational Methods 8.4 Modeling and ApplicationsChapter 9. Inequality Constrained Optimization; Kuhn-Tucker Methods 9.1 Introduction 9.2 Basic Theory 9.3 Graphical Interpretation and Computational Methods 9.4 Modeling and ApplicationsChapter 10. Method of Feasible Directions and Other Special NL Methods 10.1 Methods of Feasible DirectionsNumerical methods (Directional Searches)Starting Point Methods 10.2 Separable Programming 10.3 Quadratic ProgrammingChapter 11. Dynamic Programming 11.1 Introduction 11.2 Continuous Dynamic Programming 11.3 Modeling and Applications with Continuous DP 11.4 Discrete Dynamic Programming 11.5 Modeling and Applications with Discrete Dynamic Programming

Optimization is the act of obtaining the "best" result under given circumstances. In design, construction, and maintenance of any engineering system, engineers must make technological and managerial decisions to minimize either the effort or cost required or to maximize benefits. There is no single method available for solving all optimization problems efficiently. Several optimization methods have been developed for different types of problems. The optimum-seeking methods are mathematical programming techniques (specifically, nonlinear programming techniques). Nonlinear Optimization: Models and Applications presents the concepts in several ways to foster understanding. Geometric interpretation: is used to re-enforce the concepts and to foster understanding of the mathematical procedures. The student sees that many problems can be analyzed, and approximate solutions found before analytical solutions techniques are applied. Numerical approximations: early on, the student is exposed to numerical techniques. These numerical procedures are algorithmic and iterative. Worksheets are provided in Excel, MATLAB®, and Maple" to facilitate the procedure. Algorithms: all algorithms are provided with a step-by-step format. Examples follow the summary to illustrate its use and application. Nonlinear Optimization: Models and Applications: Emphasizes process and interpretation throughout Presents a general classification of optimization problems Addresses situations that lead to models illustrating many types of optimization problems Emphasizes model formulations Addresses a special class of problems that can be solved using only elementary calculus Emphasizes model solution and model sensitivity analysis About the author: William P. Fox is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School. He received his Ph.D. at Clemson University and has taught at the United States Military Academy and at Francis Marion University where he was the chair of mathematics. He has written many publications, including over 20 books and over 150 journal articles. Currently, he is an adjunct professor in the Department of Mathematics at the College of William and Mary. He is the emeritus director of both the High School Mathematical Contest in Modeling and the Mathematical Contest in Modeling.

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