# Ralph GomoryNew York University | NYU · Leonard N. Stern School of Business

Ralph Gomory

Ph.D

## About

92

Publications

30,068

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10,073

Citations

Citations since 2017

Introduction

**Skills and Expertise**

## Publications

Publications (92)

Corner Polyhedra are a natural intermediate step between linear programming and integer programming. This paper first describes how the concept of Corner Polyhedra arose unexpectedly from a practical operations research problem, and then describes how it evolved to shed light on fundamental aspects of integer programming and to provide a great vari...

The United States from its earliest years led the world in making the corporate form of business organization widely available to entrepreneurs. Starting in the 1790s, corporations became key institutions of the American economy, contributing greatly to its remarkable growth. This essay reviews the evolution of corporations across several eras of t...

This paper examines international trade issues as vital indicators of the economic prospects of the United States and other developed economies. In particular, it challenges misuses of the doctrine of mutual gains from trade and instead argues that comparative advantage does not guarantee increases in benefits to both trading partnersâ€”especially...

Mathematician, and father of fractal geometry, who described the roughness of nature.

Later in 1957, as the end of my three-year tour of duty in the Navy was approaching, Princeton invited me to return as Higgins
Lecturer in Mathematics. I had been aWilliams undergraduate and a then a graduate student at Cambridge and Princeton while
getting my Ph.D. I had published 4 papers in non-linear differential equations, a subject to which I...

Globalization has brought about fundamental change. But, the same theory that shows that free trade is beneficial also shows that globalization can be harmful to at least one of the trading countries. Moreover, with globalization, the interests of a country and its companies may diverge. In this paper, we examine free trade in a globalizing world a...

The prospect of biological attacks is a growing strategic threat. Covert aerosol attacks inside a building are of particular concern. In the summer of 2005, the Center for Biosecurity of the University of Pittsburgh Medical Center convened a Working Group to determine what steps could be taken to reduce the risk of exposure of building occupants af...

This report documents the work during 2002 and early 2003 of the President's Council of Advisors on Science and Technology (PCAST) Panel on the Science and Technology of Combating Terrorism. Because of the urgency of the nation's efforts to rally scientific, engineering, and technological resources in the war on terrorism, the findings of the Panel...

. Any integer program may be relaxed to a group problem. We define the master cyclic group problem and several master knapsack
problems, show the relationship between the problems, and give several classes of facet-defining inequalities for each problem,
as well as a set of mappings that take facets from one type of master polyhedra to another.

. We review the necessary background on Corner Polyhedra and use this to show how knowledge about Corner Polyhedra and subadditive
functions translates into a great variety of cutting planes for general integer programming problems. Experiments are described
that indicate the dominance of a relatively small number of the facets of Corner Polyhedra....

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 introd...

This paper is an invited contribution to the 50th anniversary issue of the journalOperations Research, published by the Institute of Operations Research and Management Science (INFORMS). It describes one person's perspective on the development of computational ...

The author briefly describes the activity of the Alfred P. Sloan Foundation in support of asynchronous learning networks (ALNs) in universities, an activity that goes back to 1992. These networks can provide the essentials of the traditional learning experience electronically, delivering the same ultimate result at roughly the same cost as traditio...

In this book Ralph Gomory and William Baumol adapt classical trade models to the modern world economy. Trade today is dominated by manufactured goods, rapidly moving technology, and huge firms that benefit from economies of scale. This is very different from the largely agricultural world in which the classical theories originated. Gomory and Baumo...

We discuss linear Ricardo models with a range of parameters. We show that the exact boundary of the region of equilibria of these models is obtained by solving a simple integer programming problem. We show that there is also an exact correspondence between many of the equilibria resulting from families of linear models and the multiple equilibria o...

We discuss the classical linear model of international trade. We show that many of the results of the scale economies models carry over to the linear case, notably the critical role of share of world output, and the resulting conflict in the interests of trading partners. We use the linear one-input production functions eijlij and Cobb-Douglas util...

In the presence of scale economies a country, that is the exclusive producer of a commodity can retain its monopoly against efforts of others to enter on a small scale even if that firm has neither absolute nor comparative advantage in its production. Equilibria can violate comparative advantage, be inefficient, and yet be stable. Equilibria that v...

I should like to address the impact of economic development in technically backward countries on economic welfare in the industrialized nations, a topic of great concern to many policy makers. As countries that have been underdeveloped improve their industrial capabilities, they can become significant contributors to the total world economy. From a...

[fre] L'analyse montre une correspondance étroite entre le nombre élevé d'équilibres dans un modèle d'échange statique avec économies d'échelles et celui d'un modèle linéaire dans lequel la productivité d'un pays partenaire à l'échange augmente par rapport à celle de son partenaire. Il est montré que la croissance de la productivité dans un pays pa...

We are the first generation of Americans who think that our
children will not live substantially better than we do. This article
takes a hard look at US productivity, education, technology, and the
prospects of improving national output. The problem of national
productivity is not well understood, yet it matters enormously. It may
be that the answe...

In this paper we examine the impact of membership in Preferential Trade Agreements (PTAs) on trade between PTA members. Rather than considering the impact of PTA membership on the volume of trade we consider the impact of membership on the structure of trade. For a large sample of countries over the period 1962-2000 we find that membership in a PTA...

In this paper we examine the impact of membership in Preferential Trade Agreements (PTAs) on trade between PTA members. Rather than considering the impact of PTA membership on the volume of trade we consider the impact of membership on the structure of trade. For a large sample of countries over the period 1962-2000 we find that membership in a PTA...

This paper extends the analysis of the classical n-good two-country Ricardo model of international trade to the case where the production functions have economies of scale. It addresses the technical difficulties inherent in economies of scale by new integer programming and linear programming methods. The analysis reveals the existence of a well de...

The availability of both simple and advanced techniques can now solve traffic delays at toll booths. One low-tech approach involves the addition of gates slightly downstream or upstream of the existing toll plaza. Another approach employs the use of an existing cashier-manned gate to collect from alternative 'platoons' of cars. This has shown a 30%...

In this paper we examine the impact of membership in Preferential Trade Agreements (PTAs) on trade between PTA members. Rather than considering the impact of PTA membership on the volume of trade we consider the impact of membership on the structure of trade. For a large sample of countries over the period 1962-2000 we find that membership in a PTA...

The arrival of a new Administration in Washington raises once more the question of setting priorities for science and technology. However, this may not be quite the right question. Setting priorities can be difficult without goals. If we don't have goals, if we don't know where we are going, it's hard to have a sensible discussion about the best wa...

A model of international trade that resembles the classical Ricardo model but differs from it in admitting economies of scale in production is described.

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 e...

Events of the past 40 years have demonstrated in unforgettable ways that science, in addition to being a stimulating intellectual pursuit, has enormous practical power. This power was evident first in the stunningly sudden appearance of the atomic bomb. A second example of the practical power of science was the appearance and rapid evolution of the...

The industrial environment contains problems of practical importance. To exploit that environment, an industrial research group must accept the responsibility for forming strong and useful relationships with the rest of the company of which it is a part. Opportunities for contributing to both the industrial and academic worlds stem from the many pr...

Over the past 20 years computer technology has advanced rapidly, especially in the areas of logic and memory. Continued growth at a similar pace can be expected through and probably beyond the 1980's. The technology has already had such major effects on scientific research and engineering that it is of practical importance to try to predict its evo...

In technology development significant advances are as often the result of a series of evolutionary steps as they are of breakthroughs.
This is illustrated by the examples of the steam engine and the computer. Breakthroughs, such as the transistor, are relatively
rare, and are often the result of the introduction of new knowledge coming from a quite...

This study of r -separating sets was originally motivated by the maximum-flow-minimum-cut theorem of finite networks [1 ; 2]. In working toward a continuous version of the maximum-flow-minimum-cut theorem from the usual discrete version, one is led directly to the notion of r -connectedness of a set as defined below. This notion of r -connectedness...

Inequalities based on the integer nature of some or all of the variables are useful in almost any algorithm for integer programming. This chapter describes the methods for producing such inequalities with the help of group problems and subadditive functions. It outlines a general approach, taking the pure integer case. The chapter discusses a pure...

The group problem on the unit interval is developed, with and without continuous variables. The connection with cutting planes, or valid inequalities, is reviewed. Certain desirable properties of valid inequalities, such as minimality and extremality are developed, and the connection between valid inequalities for P(I, u
0) and P
-+(I, u
0) is deve...

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{stmar...

The economic lot size programming problem, as studied originally by A. S. Manne and later by B. P. Dzielinski, C. T. Baker and A. S. Manne, is the problem of making economic lot size, inventory and work force decisions in a multi-production process. When several thousand distinct items are involved, the large number of equations that result from th...

In earlier papers [Opns. Res. 9, 849–859 (1961), and 11, 863–888 (1963)] the one-dimensional cutting stock problem was discussed as a linear programming problem. There it was shown how the difficulty of the enormous number of columns occurring in the linear programming formulation could be overcome by solving a knapsack problem at every pivot step....

We consider a machine with a single real variable x that describes its state. Jobs J1, ', JN are to be sequenced on the machine. Each job requires a starting state A, and leaves a final state Bi. This means that Ji can be started only when x = Ai and, at the completion of the job, x = Bi. There is a cost, which may represent time or money, etc., fo...

A communication network is a set of nodes connected by arcs. Every arc has associated with it a nonnegative number called the branch capacity which indicates the maximum amount of flow that can pass through the arc. A communication network must have large enough branch capacities such that all message requirements (which can be regarded as flows of...

This paper describes a simple calculation for the assignment and transportation problems which is "dual to" the well-known Hungarian Method. While the Hungarian is a dual method, this method is primal and so gives a feasible assignment at each stage of the calculation. Bounds on the number of steps required for the assignment and transportation pro...

In this paper, the methods for stock cutting outlined in an earlier paper in this Journal [Opns Res 9, 849--859 1961] are extended and adapted to the specific full-scale paper trim problem. The paper describes a new and faster knapsack method, experiments, and formulation changes. The experiments include ones used to evaluate speed-up devices and t...

The Trim Problem has many forms, and most of this paper will be devoted to a discussion of the simplest one, the so-called onedimensional Trim Problem. Some other—and more difficult forms—will be mentioned at the end, but even the simplest case has both applications significance and technical difficulties that make it interesting. To gain some idea...

The cutting-stock problem is the problem of filling an order at minimum cost for specified numbers of lengths of material to be cut from given stock lengths of given cost. When expressed as an integer programming problem the large number of variables involved generally makes computation infeasible. This same difficulty persists when only an approxi...

In this article Gomory's 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.

The object of this paper is to study the behavior of a solution to a system of ordinary differential equations, dv/dt = F(v), v a real 3-vector, as this solution approaches a critical point P. The motion of the solution will be projected onto a unit sphere D around P, and the limit sets of the resulting projected motion will be studied. These limit...