Nonlinear optimization involves finding the best solution to a mathematical programming problem in which the objective function and constraints are not necessarily linear. This chapter examines optimization from a very general point of view. It considers both unconstrained and constrained models. Nevertheless, a thorough grasp of the subject of nonlinear optimization requires an understanding of both the mathematical foundations of optimization as well as the algorithms that have been developed for obtaining solutions. The chapter is intended to provide insights from both of these perspectives. Newton's method generally requires fewer iterations for convergence than the gradient search method because it uses a better direction of movement from one point to the next. Quadratic programming comprises an area of mathematical programming that is second only to linear programming in its broad applicability within the field of Operations Research. A classical problem that is often used to illustrate the use of the quadratic programming model is called portfolio selection.