Objective function approximations in mathematical programming

Mathematical programming applications often require an objective function to be approximated by one of simpler form so that an available computational approach can be used. An a priori bound is derived on the amount of error (suitably defined) which such an approximation can induce. This leads to a natural criterion for selecting the best approximation from any given class. We show that this criterion is equivalent for all practical purposes to the familiar Chebyshev approximation criterion. This gains access to the rich legacy on Chebyshev approximation techniques, to which we add some new methods for cases of particular interest in mathematical programming. Some results relating to post-computational bounds are also obtained.This paper was partially supported by the National Science Foundation and by the Office of Naval Research, and was the basis for a plenary lecture delivered at the IX International Symposium on Mathematical Programming in Budapest, Hungary, August 1976.     

Generalized Benders decomposition

J. F. Benders devised a clever approach for exploiting the structure of mathematical programming problems withcomplicating variables (variables which, when temporarily fixed, render the remaining optimization problem considerably more tractable). For the class of problems specifically considered by Benders, fixing the values of the complicating variables reduces the given problem to an ordinary linear program, parameterized, of course, by the value of the complicating variables vector. The algorithm he proposed for finding the optimal value of this vector employs a cutting-plane approach for building up adequate representations of (i) the extremal value of the linear program as a function of the parameterizing vector and (ii) the set of values of the parameterizing vector for which the linear program is feasible. Linear programming duality theory was employed to derive the natural families ofcuts characterizing these representations, and the parameterized linear program itself is used to generate what are usuallydeepest cuts for building up the representations.In this paper, Benders' approach is generalized to a broader class of programs in which the parametrized subproblem need no longer be a linear program. Nonlinear convex duality theory is employed to derive the natural families of cuts corresponding to those in Benders' case. The conditions under which such a generalization is possible and appropriate are examined in detail. An illustrative specialization is made to the variable factor programming problem introduced by R. Wilson, where it offers an especially attractive approach. Preliminary computational experience is given.Communicated by A. V. BalakrishnanAn earlier version was presented at the Nonlinear Programming Symposium at the University of Wisconsin sponsored by the Mathematics Research Center, US Army, May 4–6, 1970. This research was supported by the National Science Foundation under Grant No. GP-8740.     

A simple model for fast correction of the horizontal position of atokamak plasma

A virtual movement of the plasma in the radial direction was studied in order to obtain the magnitude of the vertical field applied to the Tokamak de Varennes plasma for fast horizontal motion using coils inside the vacuum vessel. The developed model includes the radial dependence of several parameters     

The design and implementation of a prototype structured modeling environment

This paper describes the design and implementation of the prototype structured modeling environment FW/SM. The underlying design principles provide the central focus. Other points of interest include discussions of FW/SM's delivery platform, its interface to external packages, and its optimization interfaces. The intended audience is all modeling system evaluators, designers, and implementors, including those who do not happen to take a structured modeling approach.