Alan J. Hoffman

May 30, 1924

Brief Biography

Alan Hoffman was born and raised in New York City and entered Columbia University on a Pulitzer scholarship in 1940 at the age of 16, intending to become a mathematics professor. He left college early to serve in the U.S. Army 1943-46, primarily with the 3186th Signal Service Battalion in Europe (England, France, Germany) and the Pacific (Philippine Islands, Japan).  While in basic training he developed some ideas, about axioms for a geometry of circles, that later became the genesis of his doctoral dissertation on the foundations of inversion geometry. 

Returning to Columbia, he received his AB degree in 1947 and PhD degree in 1950.  After a postdoctoral year at the Institute for Advanced Study in Princeton (where his office was next to Einstein's), he joined the Applied Mathematics Division of the National Bureau of Standards (NBS, now the National Institute of Standards and Technology) in Washington DC.  There he became a leader of the Bureau's research in a new subject: linear programming.  He learned linear programming from George Dantzig, worked with the U.S. Air Force Project SCOOP, and became acquainted with other operations research notables such as Richard Bellman and Harold Kuhn.  He also lectured at American University and George Washington University in DC.  At NBS he and coauthors published a paper showing, based on experiments, that the simplex method was computationally superior to its contemporary competitors.  He also developed the first example of cycling in the simplex method and wrote about totally unimodular matrices, Lipschitz conditions for systems of linear inequalities, bounds on eigenvalues of normal matrices and the properties of smooth patterns of production. He was a key organizer of the Second Symposium in Linear Programming, held at the Bureau in January 1955.

In 1956, Hoffman left the Bureau to become a Scientific Liaison Officer (mathematics) at the London branch of of the Office of Naval Research, returning to the US after eighteen months to add business to his military, government and university experiences by joining the Management Consultation Services unit of General Electric Company (GE) in New York.  His job involved teaching operations research to students from various GE departments. In 1961, he moved from GE to  IBM's T. J. Watson Research Center at Yorktown Heights, New York. 

Hoffman was a Research Staff Member in the Mathematical Sciences Department for 41 years (serving as its acting director for nearly a year), became an IBM Fellow in 1977  and retired in 2002 as an IBM Fellow Emeritus. Over the course of his career he published upwards of 200 academic papers, more than a third of them with coauthors. 

As a member of IBM's Mathematical Sciences Department, Hoffman worked closely with Ralph GomoryPhilip S. Wolfe, and many others.   While highly active in research at IBM, Hoffman also served as adjunct or visiting professor at the Technion Israel Institute of Technology (which awarded him an honorary doctorate), Yale, Stanford, Rutgers, the Georgia Institute of Technology, and the City University of New York.  Unusual for an adjunct professor, he supervised fifteen PhD students at four of these institutions.  In Who's Who in America he is listed as both mathematician and educator. 

Hoffman served on the editorial board of eleven different journals including Linear Algebra and its Applications, which he helped establish as its first editor-in-chief in 1968. In 1992, together with Wolfe he was awarded the John von Neumann Theory Prize by ORSA and TIMS, predecessors of INFORMS. In presenting the award, George Nemhauser recognized Hoffman and Wolfe as the intellectual leaders of the mathematical programming group at IBM. He cited Hoffman for his work in combinatorics and linear programming and for his early work on the computational efficiency of the simplex method during his time at NBS. 

In a biography published in an issue of Linear Algebra and its Applications dedicated to Hoffman on the occasion of his sixty-fifth birthday, Uriel Rothblum wrote that "Above and beyond his scholarly and professional contributions, Hoffman has unparalleled ability to enjoy everything he does.  He enjoys singing, ping pong, puns, witty stories, and -- possibly as much as anything else -- doing mathematics."   

Other Biographies

Wikipedia Entry for Alan Hoffman (mathematician)

Rothblum U. G. (1989) Preface. Linear Algebra and its Applications, Special Issue Dedicated to Alan J. Hoffman, 114-115: 1-16. 


Columbia University, A.B. 1947 

Columbia University, PhD 1950 (Mathematics Genealogy


Academic Affiliations
Non-Academic Affiliations

Key Interests in OR/MS


Oral Histories

Alan Hoffman Interview by Irv Lustig May 4, 2001. Video by Irv Lustig, Short Hills, NJ.

NOTE: The video chapter transcripts are searchable, with search results displayed as marks on the time bar above the search box.  Click a mark to jump to the search word or phrase in the video and transcript, or click on any word in the transcript to jump to that point in the video.


Jump to Chapters

Chapter 1: First Job at National Bureau of Standard
Chapter 2: Cycling and the Simplex Method
Chapter 3: Evolution of the Hoffman-Wielandt Theorem
Chapter 4: Duality and the König-Egerváry Theorem
Chapter 5: Computational Comparisons with the Simplex Method
Chapter 6: An Early Combinatorial Auction
Chapter 7: Solving Linear Programs on SEAC
Chapter 8: Explaining Optimization
Chapter 9: A Historical Perspective
Chapter 10: Looking Forward
Chapter 11: The Growth Of Optimization
Chapter 12: Criticism of Optimization
Chapter 13: Concluding Remarks

Memoirs and Autobiographies


Hoffman A. J. (1991) Linear Programming at the National Bureau of Standards. Lenstra J. K., Rinnoy-Kan A., & Schrijver A. in History of Mathematical Programming, a collection of personal reminiscences, 62-64. Elsevier Science Publishers: New York. 

Hoffman A. J. (2003) Autobiographical Notes, in Micchelli C. A. (2003) Selected Papers of Alan Hoffman, With Commentary. World Scientific, River Edge NJ pp. xxiii - xliv

Awards and Honors

National Academy of Science 1982

American Academy of Arts and Sciences 1987

John von Neumann Theory Prize 1992  

Founder's Award, Mathematical Programming Society 2000  

The Institute for Operations Research and the Management Sciences Fellow 2002

Selected Publications

Hoffman A. J. (1953) Cycling in the simplex algorithm, Technical report 2974. National Bureau of Standards.

Hoffman A. J. & Jacobs W. (1954) Smooth patterns of production. Management Science, 1(1): 86-91.

Hoffman A. J. & Kruskal J. B. (1956) Integral Boundary Points of Convex Polyhedra. Kuhn H. W. & Tucker A. J., eds. Linear Inequalities and Related Systems, 223-246. Princeton University Press: Princeton.  

Hoffman A. J. (1960) Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. Combinatorial Analysis: Proceedings of the Tenth Symposium in Applied Mathematics, 113-127.

Gilmore P. C. & Hoffman A. J. (1962) A characterization of comparability graphs and of interval graphs, No. RC-651. IBM Watson Research Center: Yorktown Heights, New York.

Gomory R. & Hoffman A. J. (1963) On the convergence of an integer-programming process. Naval Research Logistics Quarterly, 10(1): 369-373.

Hoffman A. J. (1963) On abstract dual linear programs. Naval Research Logistics Quarterly, 10(1): 369-373.

Barnes E. R. & Hoffman A. J. (1984) Partitioning, spectra, and linear programming. Progress in Combinatorial Optimization, 13-25. Wiley: New York.

Hoffman A. J. & Wolfe P. (1985) History. Lawler E. L., Lenstra J. K., Rinnoy Kan A. H. G., & Shmoys D. B., eds. The Traveling Salesman Problem. John Wiley & Sons: New York. 

Additional Resources

Micchelli C. A. (2003) Selected Papers of Alan Hoffman, With Commentary. World Scientific, River Edge NJ  [Note: each section begins with Hoffman's recollections of the origins of the papers in it.]