Fundamental Engineering Optimization Methods
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About the book
Reviews
David N. Murray ★★★★★
Very specific and concise. Familiarity of some concepts is indeed needed to achieve spontaneous understanding of the concepts involved and presented.
Kalle N ★★★★☆
Mycket bra innehåll, väl skriven
Description
This book is addressed to students in the fields of engineering and technology as well as practicing engineers. It covers the fundamentals of commonly used optimization methods in engineering design. These include graphical optimization, linear and nonlinear programming, numerical optimization, and discrete optimization. Engineering examples have been used to build an understanding of how these methods can be applied. The material is presented roughly at senior undergraduate level. Readers are expected to have familiarity with linear algebra and multivariable calculus.
Preface
This book is addressed to students in fields of engineering and technology as well as practicing engineers. It covers the fundamentals of commonly used optimization methods used in engineering design. Optimization methods fall among the mathematical tools typically used to solve engineering problems. It is therefore desirable that graduating students and practicing engineers are equipped with these tools and are trained to apply them to specific problems encountered in engineering practice.
Optimization is an integral part of the engineering design process. It focuses on discovering optimum solutions to a design problem through systematic consideration of alternatives, while satisfying resource and cost constraints. Many engineering problems are openended and complex. The overall design objective in these problems may be to minimize cost, to maximize profit, to streamline production, to increase process efficiency, etc. Finding an optimum solution requires a careful consideration of several alternatives that are often compared on multiple criteria.
Mathematically, the engineering design optimization problem is formulated by identifying a cost function of several optimization variables whose optimal combination results in the minimal cost. The resource and other constraints are similarly translated into mathematical relations. Once the cost function and the constraints have been correctly formulated, analytical, computational, or graphical methods may be employed to find an optimum. The challenge in complex optimization problems is finding a global minimum, which may be elusive due to the complexity and nonlinearity of the problem.
This book covers the fundamentals of optimization methods for solving engineering problems. Written by an engineer, it introduces fundamentals of mathematical optimization methods in a manner that engineers can easily understand. The treatment of the topics presented here is both selective and concise. The material is presented roughly at senior undergraduate level. Readers are expected to have familiarity with linear algebra and multivariable calculus. Background material has been reviewed in Chapter 2.
The methods covered in this book include a) analytical methods that are based on calculus of variations; b) graphical methods that are useful when minimizing functions involving a small number of variables; and c) iterative methods that are computer friendly, yet require a good understanding of the problem. Both linear and nonlinear methods are covered. Where necessary, engineering examples have been used to build an understanding of how these methods can be applied. Though not written as text, it may be used as text if supplemented by additional reading and exercise problems from the references.
There are many good references available on the topic of optimization methods. A short list of prominent books and internet resources appears in the reference section. The following references are main sources for this manuscript and the topics covered therein Arora (2012); Belegundu and Chandrupatla (2012); Chong and Zak (2013); and, Griva, Nash & Sofer (2009). In addition, lecture notes of eminent professors who have regularly taught optimization classes are available on the internet. For details, the interested reader may refer to these references or other web resources on the topic.
Content
Preface
 Engineering Design Optimization
 Introduction
 Optimization Examples in Science and Engineering
 Notation
 Mathematical Preliminaries
 Set Definitions
 Function Definitions
 Taylor Series Approximation
 Gradient Vector and Hessian Matrix
 Convex Optimization Problems
 Vector and Matrix Norms
 Matrix Eigenvalues and Singular Values
 Quadratic Function Forms
 Linear Systems of Equations
 Linear Diophantine System of Equations
 Condition Number and Convergence Rates
 ConjugateGradient Method for Linear Equations
 Newton’s Method for Nonlinear Equations
 Graphical Optimization
 Functional Minimization in OneDimension
 Graphical Optimization in TwoDimensions
 Mathematical Optimization
 The Optimization Problem
 Optimality criteria for the Unconstrained Problems
 Optimality Criteria for the Constrained Problems
 Optimality Criteria for General Optimization Problems
 Postoptimality Analysis
 Lagrangian Duality
 Linear Programming Methods
 The Standard LP Problem
 The Basic Solution to the LP Problem
 The Simplex Method
 Postoptimality Analysis
 Duality Theory for the LP Problems
 NonSimplex Methods for Solving LP Problems
 Optimality Conditions for LP Problems
 The Quadratic Programming Problem
 The Linear Complementary Problem
 Discrete Optimization
 Discrete Optimization Problems
 Solution Approaches to Discrete Problems
 Linear Programming Problems with Integral Coefficients
 Integer Programming Problems
 Numerical Optimization Methods
 The Iterative Method
 Computer Methods for Solving the Line Search Problem
 Computer Methods for Finding the Search Direction
 Computer Methods for Solving the Constrained Problems
 Sequential Linear Programming
 Sequential Quadratic Programming
References