Ralph E. Gomory

The Theory and Computation of Knapsack Functions

In earlier papers on the cutting stock problem we indicated the desirability of developing fast methods for computing knapsack functions. A one-dimensional knapsack function is defined by: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}

Some continuous functions related to corner polyhedra, II

f(x)= \max \{\Pi_{1} Z_{1} + \cdots + \Pi_{m}Z_{m};\enspace l_{1}Z_{1} + \cdots + l_{M}Z_{m}\leq x,\; Z_{i}\geq 0,\; Z_{i}\ \mbox{integer}\}

Science and Product.

\end{document} where Πi and li are given constants, i = 1, …, m. Two-dimensional knapsack functions can also be defined. In this paper we give a characterization of knapsack functions and then use the characterization to develop more efficient methods of computation. For one-dimensional knapsack functions we describe certain periodic properties and give computational results.

T-space and cutting planes

Abstract This paper first describes a theory and algorithms for asymptotic integer programs. Next, a class of polyhedra is introduced. The vertices of these polyhedra provide solutions to the asymptotic integer programming problem; their faces are cutting planes for the general integer programming problem and, to some extent, the polyhedra coincide with the convex hull of the integer points satisfying a linear programming problem. These polyhedra are next shown to be cross sections of more symmetric higher dimensional polyhedra whose properties are then studied. Some algorithms for integer programming, based on a knowledge of the polyhedra, are outlined.

Corner Polyhedra and their connection with cutting planes

Previous work on Gomorys corner polyhedra is extended to generate valid inequalities for any mixed integer program. The theory of a corresponding asymptotic problem is developed. It is shown how faces previously generated and those given here can be used to give valid inequalities for any integer program.

Early Integer Programming

In this article Gomorys method of solution of integer linear programming problems is described briefly (with an example of the method of solution). The bulk of the paper is devoted to a discussion of the dual prices and their relationship to the marginal yields of scarce indivisible resources and their efficient allocation. IT HAS been known for some time that a method of solution of the general linear programming problem in which the variables are required to take integer values would also permit the solution of a considerable variety of other problems many of which are not obviously related to it.1 For example, Markowitz and Manne [13] have shown that the difficult concave (nonlinear) programming problem (e.g., a cost minimization problem in which the total cost function is shaped like a hill) can, at least in principle, be approximated as an integer program which permits the determination of a global, and not just a local minimum. Nonconvex feasible regions can also, at least in principle, be handled by integer programming. Among the economic problems which are related to integer programming are the travelling salesman problem and problems in which fixed (inescapable) costs are present. A surprisingly wide range of problems including diophantine problems and the four color map problem2 can be given an integer programming formulation. Some of these applications will be described in greater detail in section five of this paper. Recently one of the authors of this article developed a method, which he calls the method of integer forms (MIF), for solving integer programming problems. In the next section the method of solution will be described in some detail. No proof that the algorithm arrives at the optimal integer solution in a finite number of steps will be described since it is rather lengthy and is being published elsewhere (see Gomory [6] and [7]. For an alternative approach see Land and Doig [12]).

TRAJECTORIES TENDING TO A CRITICAL POINT IN 3-SPACE*

Much of what needs to be changed in U.S. industry involves close ties to manufacturing, design for manufacturability, a rapid design cycle, and up-to-date technical knowledge on the part of the engineers themselves. Being up-to-date requires conscious company effort. Traveling to meetings, reading the technical literature, and being a part of the engineering community are necessities if we are to compete with others who make these efforts and are thus better able to incorporate technical change rapidly into their own products. Outside the product improvement cycle, a research (as opposed to development) organization in industry must have close ties to development and manufacturing in order to succeed. With these close ties, researchers can understand the progress of the cycle and can introduce new steps at the appropriate time and in an acceptable form. A research organization that surmounts the internal barriers and becomes an accepted contributor to the development and manufacturing process can, because of its greater technical depth, its scientific knowledge, and its close ties with the university world, become a forceful initiator of progress. It is more difficult, in our opinion, to make these contributions from a university base and from government laboratories as they are now constituted. Much has been said by industry and government leaders about reforming the educational system and strengthening the national science base—things that help build a strong foundation. A strong science base supplies a vast storehouse of new ideas, and a good educational system provides engineers and manufacturing workers with knowledge; but strength here cannot make up for inadequacies in the functioning of the development and manufacturing cycle. The United States must learn to succeed, not only in the ladder type of innovation in which a wholly new idea from science creates a wholly new product (the science-dominated process at which we have succeeded in the past), but also in the rapid, cyclical, engineer-dominated process of incremental product improvement. Neither process is a substitute for the other; we need both.

Computers in Science and Technology: Early Indications

Abstract. In this paper we show how knowledge about T-space translates directly into cutting planes for general integer programming problems. After providing background on Corner Polyhedra and on T-space, this paper examines T-space in some detail. It gives a variety of constructions for T-space facets, all of which translate into cutting planes, and introduces continuous families of facets. In view of the great variety of possible facets, no one of which can be dominated either by any other or by any combination of the others, a measure of merit is introduced to provide guidance on their usefulness. T-spaces based on higher dimensional groups are discussed briefly as is the idea of going beyond cutting planes to iterated approximations of Corner Polyhedra.