George B. Dantzig

Linear Programming Under Uncertainty

(This article originally appeared in Management Science, April-July 1955, Volume 1, Numbers 3 and 4, pp. 197-206, published by The Institute of Management Sciences.)

Complementary pivot theory of mathematical programming

Abstract : Problems of the form: Find w and z satisfying w = q + Mz, w = or 0, z = or 0, zw = 0 play a fundamental role in mathematical programming. This paper describes the role of such problems in linear programming, quadratic programming and bimatrix game theory and reviews the computational procedures of Lemke and Howson, Lemke, and Dantzig and Cottle.

Chemical Equilibrium in Complex Mixtures

A new method for the determination of the equilibrium composition of complex mixtures is described. The general method, which is based on the minimization of free energy, states the problem with unusual simplicity, avoiding many of the usual difficulties of description and computation. Two specific computation procedures are shown, one using a steepest descent technique applied to a quadratic fit, the other making use of linear programing.

Linear inequalities and related systems

The description for this book, Linear Inequalities and Related Systems. (AM-38), will be forthcoming.

ON THE CONTINUITY OF THE MINIMUM SET OF A CONTINUOUS FUNCTION

Abstract : This memorandum obtains necessary and sufficient conditions so that the solution of a constrained minimization problem will vary continuously when the constraints and objective function are varied. It also obtains special results in the case that the constraints are linear inequalities.

Multi-stage stochastic linear programs for portfolio optimization

The paper demonstrates how multi-period portfolio optimization problems can be efficiently solved as multi-stage stochastic linear programs. A scheme based on a blending of classical Benders decomposition techniques and a special technique, called importance sampling, is used to solve this general class of multi-stochastic linear programs. We discuss the case where stochastic parameters are dependent within a period as well as between periods. Initial computational results are presented.

Parallel processors for planning under uncertainty

Our goal is to demonstrate for an important class of multistage stochastic models that three techniques — namely nested decomposition, Monte Carlo importance sampling, and parallel computing — can be effectively combined to solve this fundamental problem of large-scale linear programming.

ON THE SIGNIFICANCE OF SOLVING LINEAR PROGRAMMING PROBLEMS WITH SOME INTEGER VARIABLES

Abstract : Recent proposals by Gomory and others for solving linear programs involving integer-valued variables appear sufficiently promising that it is worthwhile to systematically review and classify problems that can be reduced to this class and thereby solved. Historically, non-linear, nonconvex and combinatorial problems are areas where classical mathematics almost always fails. It is therefore significant that the reduction can be made for problems involving multiple dichotomies and k-fold alternatives which include problems with discrete variables, non-linear separable minimizing functions, conditional constraints, global minimum of general concave functions and combinatorial problems such as the fixed charge problem, traveling salesman problem, orthogonal latin square problems, and map coloring problems.

Letter to the Editor---A Comment on Edie's “Traffic Delays at Toll Booths”

E. W. Paxson of Rand has suggested that linear programming methods could be used as an alternative procedure in the scheduling section of Leslie C. Edies interesting paper on “Traffic Delays at Toll Booths” (Leslie C. Edie. 1954. Traffic delays at toll booths. J. Opns. Res. Soc. Am. 2 107.). The purpose of this note is to elaborate on this suggestion. Operations Research, ISSN 0030-364X, was published as Journal of the Operations Research Society of America from 1952 to 1955 under ISSN 0096-3984.

A generalization of the linear complementarity problem

Abstract The linear complementarity problem: find z∈Rp satisfying w = q + M z w ⩾ 0 , z ⩾ 0 ( LCP ) z T w = 0 is generalized to a problem in which the matrix M is not square. A solution technique similar to C. E. Lemkes (1965) method for solving (LCP) is given. The method is discussed from a graph-theoretic viewpoint and closely parallels a proof of Sperners lemma by D. I. A. Cohen (1967) and some work of H. Scarf (1967) on approximating fixed points of a continuous mapping of a simplex into itself.