Abstract
The purpose of this paper is to describe the multiplier method for finding the solution of constrained nonlinear programming problems. The method is based on transforming a given constrained problem into a sequence of unconstrained problems by defining the multiplier function.
The algorithm proposed is to determine unconstrained minima of the multiplier function for fixed parameters and some Lagrange multipliers. The Lagrange multipliers are then varied by a simple correction and the multiplier function is minimized again. It is proved that the multiplier method is locally convergent to the saddle points of the multiplier function at the rate of a geometric progression by proceeding in this fashion.
The features of the multiplier, function are that it is a class of the generalized Lagrangian and is twice continuously differentiable at the boundary of the feasible region. The method deals with non-convex programming problems and determines the Lagrange multipliers as well as the solution of the original problem. The numerical difficulties encountered with existing penalty methods are avoided. Numerical examples are presented to illustrate the typical convergence characteristics of the method.
