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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.

This article is based on an idea proposed by C. W. Carroll for transforming a mathematical programming problem into a sequence of unconstrained minimization problems. It describes the theoretical validation of Carroll's proposal for the convex programming problem. A number of important new results are derived that were not originally envisaged: The method generates primal-feasible and dual-feasible points, the primal objective is monotonically decreased, and a subproblem of the original programming problem is solved with each unconstrained minimization. Briefly surveyed is computational experience with a newly developed algorithm that makes the technique competitive with known methodology. (A subsequent article describing the computational algorithm is in preparation.)

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