Computable optimal value bounds for generalized convex programs

It has been shown by Fiacco that convexity or concavity of the optimal value of a parametric nonlinear programming problem can readily be exploited to calculate global parametric upper and lower bounds on the optimal value function. The approach is attractive because it involves manipulation of information normally required to characterize solution optimality. A procedure is briefly described for calculating and improving the bounds as well as its extensions to generalized convex and concave functions. Several areas of applications are also indicated

The Sequential Unconstrained Minimization Technique for Nonlinear Programing, a Primal-Dual Method

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.)

Extensions of SUMT for Nonlinear Programming: Equality Constraints and Extrapolation

This paper extends the Sequential Unconstrained Minimization Technique for nonlinear programming to include problems where the constraints are a mixture of inequalities and equalities. Theorems indicating the dual nature of the method and the solution of a subproblem are given. Finally, the theoretical basis for a k-point extrapolation procedure is established.

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.