Chapter 1 - Introduction

Abstract

The equality-constrained problem and its inequality-constrained version are the two nonlinear programming problems. Computational methods for solving these problems became the subject of intensive investigation during the late fifties and early sixties. This chapter discusses three approaches that were pursued. The first approach was based on the idea of iterative descent within the confines of the constraint set. Feasible direction methods by their nature were unable to handle problems with nonlinear equality constraints, and some of them were inapplicable or not suited for handling nonlinear inequality constraints as well. Many modifications were proposed for treating nonlinear equality constraints, but these involved considerable complexity and detracted from the appeal of the descent idea. The second approach was based on the possibility of solving the system of equations and inequalities that constitute necessary conditions for optimality for the optimization problem. The third approach was based on the elimination of constraints through the use of penalty functions.