Chapter 25 - Methods of nonlinear programming

Abstract

This chapter focuses on the methods of solution for general nonlinear programs. Several methods have been developed where either f or all gi are nonlinear or only f is nonlinear. The chapter states various methods including separable programming, Kelley's cutting plane algorithm, Zoutendijk's method of feasible direction, Rosen's Gradient projection method, Wolfe's reduced gradient method, Zangwill's Convex–Simplex Method, and Dantzig's method for convex programming. Separable programming is a special technique for obtaining solutions of a class of nonlinear programming problems where the functions involved can be expressed as a sum of functions each of a single variable only—that is, the functions f(X) and gi(X) are of the form. The linear function and the nonlinear function are examples of separable function. There are some functions, which are not directly separable can also be made so by transformation of variables. To be able to make use of the simplex method for finding a solution of the problem, the separable functions fj(xj) and gij(xj), i = 1,2..m are first approximated by piecewise linear functions.