An introduction to nonlinear programming—IV Numerical methods for constrained minimization

Abstract

In this paper, we consider the problem of determining the numerical solution of constrained minimization problems. This discussion complements the theoretical development regarding the nature of the solution of the general nonlinear programming problem that was presented in Part I of this series of articles. As in our earlier discussion, the objective of this article is to review basic ideas and to illustrate the application of the ideas by describing specific computational algorithms. Thus, we discuss a variety of algorithms but not always in their greatest detail. References are provided for more detailed expositions and for generalizations and extensions of the basic algorithms. Attention is given first to the solution of problems with linear constraints. Then, approximation methods that reduce nonlinearly constrained problems to a sequence of linear programming problems are described. Finally, the discussion is completed by describing methods that reduce nonlinearly constrained problems to a sequence of unconstrained problems. Much of the discussion of specific algorithms draws upon the results presented in Parts II and III of this series.