Numerical Experiment with the Introduction of a Scaling Factor to the Multiplier Method to Solve Constrained Optimization Problems

This paper discusses the intertwining of the Conjugate Gradient Method (CGM) as a scaling factor for Newton’s algorithm, which is employed to determine the solution of the constrained optimization problems. The constrained problem has to be converted to an unconstrained problem via the multiplier method, after which a scaling is introduced to Conjugate Gradient Method (CGM) of Hestenes and Stiefel that forces the problem to collapse toward the resulting equation quickly. Newton’s algorithm, which was embedded in the algorithm of the multiplier method, was used to carry out the minimization process. At each descent direction search, a formulated dynamic is used to update the Lagrange multiplier, such that this is done at each one-dimensional search until the optimality is reached. The use of this method was not limited to optimization problems alone, it has been extended to optimal control problems of the Lagrange and the Meyer forms, with a huge success recorded when the results obtained were compared with results of other existing methods.