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. The methods covered in this book include: analytical methods that are based on calculus of variations; graphical methods that are useful when minimizing functions involving a small number of variables; and iterative methods that are computer friendly, yet require a good understanding of the problem. Both linear and nonlinear methods are covered. 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. Contents: 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. Conjugate-Gradient Method for Linear Equations. Newton’s Method for Nonlinear Equations. Graphical Optimization. Functional Minimization in One-Dimension. Graphical Optimization in Two-Dimensions. 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. Non-Simplex 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.