The Use of Constraint Management Techniques in the Generalized Reduced Gradient Method

Abstract

Abstract The Generalized Reduced Gradient (GRG) method has proven to be an effective strategy for solving constrained nonlinear programming problems, and this paper proposes the use of constraint management techniques to enhance the GRG method. Some of the troublesome issues in the method such as the selection of the initial basis and the problem of basis interchange can be resolved if the constraints are represented in the form of a nonlinear occurrence matrix. A heuristic algorithm is presented to determine an initial partition of the state and decision variables. The procedure is aimed at minimizing the nonlinear component in the set of state variables so that computational effort during the numerical iterations is reduced. Another algorithm is presented to automate the basis interchange by using a linearized model of the governing equations and a backward dependency procedure. An improved canonical form for the general nonlinear programming problem that has the objective function embedded in the set of equality constraints is also presented. Constraint management ideas are also used to decompose the basis Jacobian matrix into smaller irreducible subsets. Several mathematical and design problems are posed as test cases for the Constraint Management Based Generalized Reduced Gradient (CMB-GRG) procedure, and representative results are presented.