Computational Algorithm for the Sequential Unconstrained Minimization Technique for Nonlinear Programming

In a previous article [Fiacco, A. V., G. P. McCormick. 1964. The sequential unconstrained minimization technique for nonlinear programming, a primal-dual method. Management Sci. 10(2) 360-366.] the authors gave the theoretical validation of the sequential unconstrained minimization technique for solving the convex programming problem. The technique is based on an idea proposed by C. W. Carroll [Carroll, C. W. 1961. The created response surface technique for optimizing nonlinear restrained systems. Oper. Res. 9(2) 169-184; Carroll, C. W. 1959. An operations research approach to the economic optimization of a Kraft Pulping Process. Doctoral dissertation, The Institute of Paper Chemistry, Appleton, Wisc.]. The method has been implemented via an algorithm based on a second-order gradient technique that has proved extremely efficient on a considerable number of test problems of varying complexity. This paper explores the computational aspects of the method. Included are discussions of parameter selection, convergence criteria, and methods of minimizing an unconstrained function. It is shown that the problem variables, on the trajectory of minima of the sequence of unconstrained functions, can be developed as functions of a single parameter. This provides the theoretical basis for an extrapolation technique that significantly accelerates convergence in actual computations. The detailed computer solution of a small example is given to illustrate the typical convergence characteristics of the method. The speed and accuracy of the computational procedure are believed to be competitive with other known techniques for solving the convex programming problem.