ABSTRACT
This chapter provides a brief discussion of optimization in the case of the finite dimensional problem. It begins with the free or unconstrained problem and shows that with enough smoothness on objective function, a "complete" set of necessary and sufficient conditions can be obtained by using the Taylor series expansion. The chapter considers the equality constrained problem and shows that the similar, though more complicated, necessary and sufficient conditions can be obtained. It obtains necessary and sufficient conditions for the inequality constrained problem and introduces Newton's method, which is the basic numerical tool in solving nonlinear equations. The chapter considers the problem of equality and inequality constraints defined. The beginning material is standard in modern day nonlinear programming texts and is based upon the idea of active constraints considered. The chapter shows that by adding extra variables it is easy to reformulate inequality constrained problems as equality constrained problems.
