This book is intended as a text covering the central concepts of penalty and barrier function methods for constrained optimization problems. The lucid presentation of the text provides a good understanding of the sequential penalty/or barrier methods and the exact penalty methods. As the field of optimization, the book should be useful to operations researchers, engineers, economists, management scientists, and other specialists from the host of disciplines from which practical optimization application?s are drawn. Algorithms and MATLAB codes are developed for both penalty and barrier methods and problems that have appeared frequently in the optimization literature are solved using the developed MATLAB code. The exact penalty functions: the exact absolute value and the augmented Lagrangian penalty function (ALPF) are also discussed in detail. In this case a single application of an unconstrained minimization technique as against to the sequential methods is used to solve the constrained optimization problems.