ABSTRACTS SAMENVATTINGEN
Visit Tinne Hoff Kjeldsen

April, 19-23.



From April 19 until April 23 dr. Tinne Hoff Kjeldsen will be the guest of the mathematics department of Nijmegen University. During her stay in the Netherlands she will give two lectures in several places. Both talks will be on the history of Nonlinear Programming (abstracts below). The talks are planned as follows.



Programme

April 21, 10:30-11:30. Nijmegen University, Department of Mathematics Staff Colloquium. Room CK N4. The Kuhn-Tucker Theorem in Nonlinear Programming: a multiple discovery?

April 22, 15:00-16:00. Utrecht University, Institute for History and Philosophy of Science. Staff Colloquium. The Kuhn-Tucker Theorem in Nonlinear Programming: a multiple discovery?

April 23, 16:00-17:00. Amsterdam Free University (VU), Department of Mathematics Staff Colloquium. The Origin of Nonlinear Programming




The Kuhn-Tucker Theorem in Nonlinear Programming: a multiple discovery?

Abstract
It was the so-called Kuhn-Tucker Theorem that launced the theory of nonlinear programming in 1950. This theorem gives necessary and sufficient conditions for the existence of an optimal solution to a nonlinear programme. The theorem was proved in a paper Nonlinear Programming written jointly by the two Princeton mathematicians Albert W. Tucker and Harold W. Kuhn. Later it turned out that this theorem had been proved already, even twice: First in 1939 in a masters thesis by William Karush from the University of Chicago. This result was never published. Second by Fritz John in a paper that was rejected by the Duke Mathematics Journal, but later published in a collection of essays for the Courant Anniversary volume in 1948. In the talk I will adress the question why the three versions of the result now known as the Kuhn-Tucker theorem had so different impact in the history of mathematics. I will discus the question of Kuhn-Tuckers theorem as a multiple discovery on the basis of a contextualized historical analysis of the works of Karush, John and Kuhn and Tucker.



The Origin of Nonlinear Programming

Abstract
The beginning of the mathematical theory of nonlinear programming can be dated back to the important paper Nonlinear programming by Albert W. Tucker and Harold W. Kuhn from 1950. Their work grew out of a project on game theory and linear programming initiated after the second world war by the Office of Naval Research in The United States. In the talk I will concentrate on the early history of the mathematical theory of nonlinear programming. It will be seen that even though nonlinear programming originated in a context of linear programming the driving forces behind Kuhn and Tucker's development of nonlinear programming was indeed very different from the stimulus that started the development of linear programming. Based on historical studies I will argue that the duality theorem in linear programming was crucial for Kuhn and Tucker's development of nonlinear programming. I will also discus the importance of war related research in the US during World War II and the role played by Office of Naval Research after the war for the development of nonlinear programming.



The following is the summary of the thesis by dr. Tinne Hoff Kjeldsen. The thesis itself in in Danish. Title: En kontekstualiseret matematikhistorisk analyse af ikke-lineær programmering: udviklingshistorie og multipel opdagelse (1999).

Summary in English
The present dissertation is about the history of nonlinear programming. It is divided into two parts of which the first can be considered as the history of the development of nonlinear programming and the second part as a contribution to the aspect of multiple discovery in the history of mathematics.

In the first part an overall picture of the emergence and establishment of nonlinear programming as a mathematical field for research is presented. The history of George B. Dantzig's development of linear programming after the second world war and the military influence on this development is treated. Based on this and on analysis of the work in linear and nonlinear programming of Harold W. Kuhn and Albert W. Tucker it is argued, that the emergence of the duality theorem in linear programming changed the scientific state of affairs of linear programming, from being a mathematical model for a concrete, practical problem within The U.S. Airforce to become a mathematical interesting research area. It is further argued that this change, in combination with the financial support from Office of Naval Research, was crucial for the development of nonlinear programming by Kuhn and Tucker. The influence of operations research and the military on the establishment of nonlinear programming as a mathematical discipline is analysed and discussed. The history behind the emergence of the duality theorem in linear programming is examined and in connection with that the development of John von Neumann's understanding of the minimax theorem in two-person zero-sum games and the different context it appeared in from 1928 until 1944 is analysed. It is argued, that in 1928, where von Neumann's first proof of the minimax theorem was published, he did not have insight into the connections between the minimax theorem and fixpoint theorems and linear inequality systems, as it is stated in the secondary literature. This insight developed gradually from 1928 until 1944. The first part also contains an analysis of the game theoretical work of Émile Borel, because it connects, through a priority debate raised by Fréchet, to a discussion about the significance of the technical context for the appreciation and propagation of a mathematical result. This discussion is an important issue in the second part of the thesis.

In the second part of the thesis the aspect of multiple discovery of the Kuhn-Tucker theorem in nonlinear programming is analysed. This aspect turns up in the history of nonlinear programming, because it is stated in the secondary literature, that the Kuhn-Tucker theorem was independently discovered by Ostrogradsky and Farkas at the end of the nineteenth century in connection with inequality constrained equilibrium in mechanics, and in modern time by W. Karush in 1939 and F. John in 1948. Based on historical analyses of the works of Ostrogradsky and Farkas, I conclude, that their results can be seen as versions of the Kuhn-Tucker theorem only if there work is interpreted within a frame work of modern physics. Today the Kuhn-Tucker theorem is ascribed also to Karush and partly to John. The historical facts is that Karush's work was a master's thesis, that were never published, Johns work was first rejected by the Duke Mathematics Journal, but then got published in a collection of essays for Courant's 60th birthday. My purpose of including this aspect of multiple discovery was to understand, why a mathematical result at a specific time, and in a specific context, can generate a new mathematical research area, while a similar result, developed almost at the same time but in a different context, did not cause any reaction. My analysis of the work of Karush, John, Kuhn and Tucker and the contexts they appeared in, give rise to the conclusion, that the technical context -and in some cases also the sociological context- had significant influence on the importance of the different results.