Evelyn Martin Lansdowne Beale Obituary

EVELYN MARTIN LANSDOWNE BEALE

8 September 1928-23 December 1985

Elected F.R.S. 1979

BY M. J. D. POWELL, F.R.S.

INTRODUCTION

EVELYN MARTIN LANSDOWNE BEALE was a talented mathematician at school and university. He became a pioneer in the development of linear programming methods at the Admiralty Research Laboratory (A.R.L.), Teddington. He then joined the Corporation for Economic and Industrial Research (C.E.I.R.) in 1961 in response to the challenge of applying operational research and mathematical programming to industrial problems. C.E.I.R. became Scicon (Scientific Control Systems Ltd) but Martin remained there, being its 'Scientific Adviser' finally, a title that reflected his strong preference for advancing his subject in a benevolent way despite the commercial pressures of industry. Regularly on Mondays from 1967 he attended the Mathematics Department at Imperial College as a visiting professor. There, at conferences and in his published work, he communicated his extraordinary skill at extracting useful results computationally from mathematical models of real problems. Most of his papers on particular calculations and on particular techniques are substantial contributions to knowledge, but probably he will be remembered best for his constant and active interest in the development of mathematical programming systems for applying optimization algorithms painlessly in practice. He wrote (1961 c)* that 'The most important part of operational research is educated common sense, and computers have absolutely no common sense', but he planned his systems so well that this defect of computers was negligible. There are no secrets of his success as he believed in open publication of useful discoveries. In all ways he was generous and kind, subject to high standards of honesty and academic integrity. He was devoted to his family and to the Christian faith.

* Numbers given in this form refer to entries in the bibliography at the end of the text.

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Biographical Memoirs

FAMILY LIFE

Martin was born on 8 September 1928, at Stanwell Moor in Middlesex, the elder child of Evelyn Stewart Lansdowne Beale and Muriel Rebecca, nee Slade. Then his father, who had studied X-rays and crystal structure with Sir William Bragg, F.R.S., was Senior Physicist and Research Engineer at the research station of the Anglo-Persian Oil Company at Sunbury. This work took the family to Persia for a year in 1932, about one year after the birth of Martin's brother Julian. On their return they moved to Markham Square, Chelsea, keeping a house in Middlesex, but Martin was unwell. Eventually it was found that he had contracted malaria in Persia, so he did not begin his formal education until 1934, at the Montessori School in London, which he attended for two years. His father left the oil business to set up a private consultancy firm for industry with R. Denman, and worked brilliantly on a wide range of engineering problems (The Times 1972). Throughout World War II his mother was Chair of the London and Middlesex branch of the Women's Land Army.

At school Martin showed an aptitude for figures, but his writing began curiously as he had learnt the alphabet from wooden bricks when 2, so to him the orientation of the letters was unimportant. He studied Latin with his mother when 8, anticipating entry to St Aubyn's preparatory school (Rottingdean, Sussex) in 1937, but he was at home throughout 1939-40 because of ill health. Home was now above Treyarnon Bay, for the Beales had become captivated by this part of Cornwall during a holiday at Constantine, and they had built Windhover House with the Denmans, an addition to their properties in Markham Square and in Middlesex. Martin studied at home with Julian and some other children during that year, and this small school continued throughout the war, although Martin returned to St Aubyn's in 1940, which had evacuated to Bettws- y-Coed. There he aimed for a scholarship to Winchester in 1941, but measles prevented him from travelling for the examination. After a further year at St Aubyn's his mathematics master, the late E. Webber, said to his wife: 'I cannot teach any more mathematics to Martin Beale, I have taught him all I know.' Martin was at Winchester from 1942 to 1946, having gained the second scholarship in 1942. Although he was the joint winner of the Richardson prize for mathematics there, he was advised not to try to follow his father from Winchester to Trinity College, Cambridge, because the competition might be too strong. Martin rejected this advice and went up to Trinity with a scholarship in 1946.

Bridge was a very strong interest of Martin's at Cambridge. His introduction to this game was a fascinating book on evaluating the strengths of hands by counting losers, which he had found when visiting his maternal grandparents at Stokenchurch in 1939. Earlier he and Julian had spent much time on devising the structure and running of an

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Evelyn Martin Lansdowne Beale

imaginary country but bridge replaced this activity. Hands were discussed on journeys to Cornwall, and he devised a new bidding system for the family, the usual four being Martin, Julian, Mrs Beale and her mother. Both Martin and Julian played for Cambridge against Oxford, and Julian went on to play for England in international matches. Having heard that Martin would work in his group at the Admiralty Research Laboratory, S. Vajda visited him at Cambridge, but apparently he showed more interest in a book he had begun to write on bidding systems (which was not published) than in his future employment. Martin did, however, join the mathematics group of the A.R.L. from Cambridge, having gained a first-class honours degree in mathematics in 1949 and a distinction in the statistics diploma in 1950.

He spent the university vacations at home in Stanwell Moor or Treyarnon Bay, and continued to live with his parents when he joined the A.R.L., moving from Stanwell Moor to Whitehall, Wraysbury (also in Middlesex) in 1952. On 1 July 1953 he married Violette Elizabeth Anne Lewis (Betty) at St Mary's Church, Hampton. They had met in time to celebrate his 22nd birthday together for she was a scientific assistant in the A.R.L. Mathematics Group. This celebration was because she had been sad to hear that Martin had spent his 21st birthday playing bridge, but he soon encouraged her to play too. An early example of his concern for teaching others occurred on a train from Waterloo when Betty sought refuge in a 'ladies only' compartment as she did not wish to receive instruction, but Martin, from the corridor, was determined to enlighten her on the lead to defeat a contract that had occurred earlier in the evening. She soon became part of the Beale family, visiting Windhover several times before their marriage, and enjoying cycling, walking and bridge with Martin. They became firm friends and dependent on each other for the rest of his life.

Their first home was a small terrace house at Strawberry Hill, Twickenham, within cycling distance of A.R.L., where Betty continued working until a few weeks before the birth of their elder son, Nicholas, in February 1955. Their daughter, Rachel, was born in May 1957. The family spent 1958 in the U.S.A. (at Princeton except for a six-week summer visit to Los Angeles), which not only helped Martin's career but also provided initial contacts with several Americans who became intimate friends of the family, including George Dantzig and Philip Wolfe. On returning to England they were able to afford a car. Their younger son, Marcus, was born in August 1960. In 1963 they moved to Ember Lane, Esher, Surrey, from 3 bedrooms to 5, Betty having found the house as Martin was so immersed in his work.

At their new home Martin would work on the sitting room floor, and even there he enjoyed the company of his children provided they did not disturb his papers. Many visitors came to Ember Lane, and E. Hellerman (Dantzig & Tomlin 1987) writes: 'One evening, I think it was in 1967, my

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wife and I were guests at the Beale home in Esher. After a fine dinner and stimulating conversation, Martin, knowing that I had a background in music, decided that he and his family would sing a Gilbert and Sullivan operetta. Each member of the family was assigned one or more roles to sing, and then to the music of a recording the performance ensued. Needless to say, I had never heard a rendition of Gilbert and Sullivan with more spirit and gusto. Martin's enthusiasm for the music spilled over into the family.' This enthusiasm included composing 'Ping went the bell' for the family and piano playing, especially the music of Haydn. Later Rachel studied music at the Dartington College of Arts, and Nicholas and Marcus too became stronger musicians than Martin. Remarkably, Nicholas and Marcus also obtained scholarships from St Aubyn's to Winchester, and then went up to Trinity College, Cambridge, where they read mathematics and architecture respectively.

On Sundays the family regularly attended church at Weston Green, near Esher, and then spent the rest of the day at Whitehall. There they thoroughly enjoyed the creations of Martin's father, including a go-cart and 'H.M.S. Octagon', which was a mock submarine conning-tower in the garden that included an escape slide and telephone communication with the house, but Martin did not inherit any engineering skills. They swam in the swimming pool there, even on Christmas Day. Every summer they visited Windhover, and there are photographs of massive systems of sandcastles that the family constructed at Treyarnon Bay.

After Martin's appointment at Imperial College, Rachel, when writing home from school, addressed her letters to 'Professor and Mrs Beale', but the school office had been advised to write to 'Mr and Mrs Beale'. When questioned on this anomaly, Rachel explained that her father was a professor only on Mondays. From then on the family adopted this distinction gleefully, and Betty recalls: 'When he became an F.R.S. everyone was being announced as they greeted the President at a Con- versazione:- " Lord and Lady X", "His Excellency the Ambassador for Y", "Professor and Mrs Z",...; Martin gave our name as "Mr and Mrs Martin Beale". I said later "You might have said Professor on this occasion". "Why? " he replied simply, " It isn't a Monday".'

His father died in 1972 and for a few years his mother continued to run both Whitehall and Windhover House, where she was visited regularly by her descendents. She then decided to retain only Whitehall, but Martin's attachment to Cornwall was so strong that he sold the house at Ember Lane in 1977 to buy Windhover. Because he needed a base near London too, Martin and his family also resided in a part of Whitehall. In a typical week he would drive to Treyarnon Bay after work on the Friday and return to Whitehall on the Sunday afternoon, much refreshed by the weekend in Cornwall. Of course he was allowed leave from his activities at Scicon, but his holidays were seldom more than two weeks long. The 1978 holiday was tragic: his brother, Julian, during one of his frequent visits to Treyarnon Bay, fell to his death in a cliff accident.

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Conferences also provided Martin with a change of scene, but he invariably attended all available lectures. Otherwise he participated enthusiastically in excursions and other activities; for example, he was among the group from the 1972 Figueira da Foz NATO Summer School who on many evenings drove furiously in bumper cars at a local fair- ground. Betty always accompanied him to the triennial symposia of the Mathematical Programming Society (M.P.S.), and there they thoroughly enjoyed meeting friends, always finding a way of offering hospitality, even at the Massachusetts Institute of Technology in 1985 where they shared a tiny room in a student dormitory. The 1973 symposium at Stanford University was unusual as Martin arrived a week early in California for a holiday. All the family travelled there, and also they all attended the meeting in Budapest in 1976, where they were in the public eye, as Martin was then Chairman of the M.P.S.

His high reputation brought him more and more commitments during the last 10 years of his life, and his response was to work even harder as he was determined not to let anyone down. After dinner he and Betty would often enjoy about 3 games of backgammon in 20 minutes and then he would spend the rest of the evening working. Immediately after a walk or a swim in Cornwall he would usually return to his studies, even during his 'holidays', but he never had a computer at home. He played intensely too, not only at board and card games, but also in the garden and on the beach, sometimes as a fierce shrieking cannibal in pursuit of his grand- children. Betty remembers that a NATO workshop in Cambridge in 1980 allowed them to be together away from home for an unusually long time: it lasted 12 days. Another visit that shows some of his personality is recalled by R. V. Simons of Scicon:

In 1984 we went to India to give a course of lectures to the Oil and Natural Gas Commission on optimization in the oil industry. From the moment that we got on the plane it was evident that he knew how to enjoy himself and to receive as well as to give. He was at home with the Indians, who were so eager to learn, and entered fully into the spirit of a coach trip around Ahmedabad and the surrounding countryside which was put on for us and for the participants on the course. Back in Delhi he showed no dismay, if he felt any, at the spartan accommodation which was provided for him at the Indian Statistical Institute and proceeded to spend his spare time sight- seeing, travelling around on the local buses rather than seeking out a more salubrious form of transport.

Usually he would wear a jacket and tie on such excursions. He was seriously ill at the beginning of 1985. Instead of a large cele-

bration of one gigasecond (109 seconds) of marriage to Betty on 9 March 1985 there was a small party. He was told at the end of August that his condition was terminal, and his response was to go on being as energetic as possible. Sir David Cox, F.R.S., of Imperial College writes: 'He was a distinguished man in a deep sense of the term. While one remembers him primarily as at the height of his activity it is impossible not to

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mention the closing months of his life. He continued working up to and beyond the limit of his strength under conditions of great discomfort, and his dignity and courage during this period will not be forgotten by those who witnessed them. In this period, as no doubt throughout his married life, he was sustained by Betty's quite exceptionally devoted care.' He was at work as usual during the last week of his life, but was taken by air to his beloved home in Cornwall on 21 December where he died on Monday, 23 December.

PROFESSIONAL CAREER

Martin's research on optimization calculations began at the Admiralty Research Laboratory in 1950 under S. Vajda who writes (Dantzig & Tomlin 1987) 'I am sometimes praised for having introduced Martin to Linear Programming. I did and I am glad of it. But there is no merit in having done it. I have introduced LP to others as well, but they did not nurture the seed the same way as Martin has done.' His early studies yielded some incisive papers that are discussed later. Also much of his time was spent on government projects. For example, K. Bowen (1986) reports that in 1951 he was the statistical adviser of the High Frequency Direction Finding Analysis Working Party, and that: 'In those days, we had to battle against Martin's unworldly knowledge, as well as battling to understand it, but we knew that we had a rare talent and we used it to the full. To instance only one contribution, we owe to Martin the operational and excellent Brooke bearing classification system.' (Details are given in references 1961 a and b.) Unpublished work at A.R.L. included studies of mine-sweeping performance and radioactive fallout. He attended the nuclear tests on Christmas Island. The 1958 calendar year was spent as a Research Associate in the Statistical Techniques Laboratory at Princeton University where his colleagues included G. E. P. Box, H. W. Kuhn, C. L. Mallows, A. Tucker and J. W. Tukey.

The mental concentration that he brought to the problems he addressed shut out conventional behaviour. Indeed K. Bowen writes (1986): 'I remember, and they will never forget, his performance for a U.S.A.- Canadian group of analysts in Room 39, in the Admiralty. With an array of tabulations and diagrams on a long table, he started by kneeling on a chair and finished on the table, wandering about the data on hands and knees.' This single mindedness continued for at least 10 years, because A. S. Douglas recalls (1986) from 1962 that: 'He never worried what people might think of him. If it improved his thought processes he would lie down and raise a leg or climb on the table or stuff his handkerchief in his mouth. Peter Windley did once comment to me that he "wished Martin would take his handkerchief out of his mouth when dropping one of his pearls of wisdom" - but it was said in amusement rather than annoyance. Indeed no one minded what he did, since he always came up with useful and perceptive remarks, from whatever position and through

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whatever obstructions they were delivered!' The quality of his early research is confirmed by S. Vajda: 'He was highly appreciated at A.R.L. When members had to be assessed, it might have been asked " Is (s)he as good as Beale?" Few were.'

C.E.I.R. (U.K.) was founded by A. S. Douglas and M. G. Kendall in 1960 as a subsidiary of C.E.I.R. in Washington. Martin joined it from A.R.L. in 1961, being no. 29 on its pay-roll. His activities there until 1964 are well described by A. S. Douglas (1986). They included modelling the diffusion equations that govern the output of oil-fields, and the simulation of air-battles for the Shape Technical Centre. He also began his very important work on the development of mathematical programming systems for the solution of general calculations, because such systems were a major part of the commercial success of the parent company in the United States. He led the British C.E.I.R. group that worked in this field

including M. Fieldhouse, R. E. Small, P. Windley and M. Jeffreys. One of the team on the American side was E. Hellerman who writes (Dantzig & Tomlin 1987) about Martin: 'His insight into how algorithms could be

implemented on a computer was phenomenal. He was the dynamo behind the extensions to the LP/90/94 System in the areas of Separable Programming, Mixed Integer, Dantzig-Wolfe Decomposition, and a host of other practical algorithms.'

Martin described these algorithms and his insight very clearly in his book Mathematical programming in practice, which was published by Pitman in 1968. His reputation was now well established, and he was much in demand as a speaker at conferences. In this year C.E.I.R. became Scicon, now a subsidiary of British Petroleum, and work had begun on the mathematical programming system UMPIRE, which superseded LP/90/94. J. A. Tomlin, who joined C.E.I.R. in 1968, writes (Dantzig & Tomlin 1987):

UMPIRE was built in an informal atmosphere which would horrify today's 'structured everything' advocates. In the first year or two there were five or six of us working for Martin on the project, which is said to have begun when Rita Walton was called into his office to write FORMC. The three or four more junior team members shared an office, and I think Martin found us a bit unruly. There was a high non-English quotient - at various times there were members from America, Australia, Canada and India, all conspiring with his German secretary - and we often found Martin's very English mannerisms amusing. I'm sure he knew this, but never, of course, referred to it. We encouraged Martin's eccentricities. When we hit a snag, or a new idea looked promising, he had a way of climbing on the nearest desk or mangling the end of his tie, while he mentally solved the problem to his own satisfaction before enlightening us. This was often much more entertaining than the work we were supposed to be doing, which Martin sometimes poured on at a fierce pace, assuming that everyone could keep up with him. One would sometimes dread those mornings when he would come in with a sheaf of papers full of new ideas, and often code to be implemented.

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Some technical details of UMPIRE and of its successor SCICONIC are given later. R. S. Hattersley of the SCICONIC team recalls: 'Martin was always refreshing to work with because he would approach any problem with an open mind, and always preferred to try a new idea than force an existing solution to fit. He produced ideas at a formidable rate without pausing for weekends, holidays, and it sometimes seemed, sleep. We were often fighting a losing battle to understand and test them all but we didn't like to disappoint him by having no results to report. Fortunately, he was always ready to stop and explain his thoughts as often as necessary, and moreover he would be genuinely concerned if he thought we were working too hard and promise to restrain his inventive- ness.' Another view of the latter part of his career at Scicon is from his secretary, B. Peberdy:

What always struck me was Martin's ability to be friendly with and make welcome eminent people or the most lowly person in the company. Although he was a splendid person to work for, he also expected people to work as hard and as efficiently as he thought them capable of. He did not suffer carelessness or fools gladly. At the same time he was very patient if people could not easily understand something. His total unstuffy, non-pompous, modest and rather humble manner belied a joy-filled and uninhibited personality, which came as a great surprise to people who didn't think past the mathematical genius to find this delightful person within. It was not uncommon to see him skipping down the corridors, skipping gleefully because he had solved an annoying mathematical problem.

In view of Martin's achievements and leadership at Scicon, it is remarkable that he also found time to make a very substantial contribution to teaching at both undergraduate and postgraduate level at Imperial College. His work there is described by J. T. Stuart, F.R.S.:

Martin Beale's association with the Department of Mathematics goes back to 1967 and he was a Visiting Professor from October of that year until his death in December 1985. Visiting Professors at Imperial College often play a very important role but I doubt if there are any who can match the regularity with which Martin came to the College and prosecuted both teaching and research work over such a long period. It is fair to say that he came on Monday of almost every week of the year during those 18 years. He taught third year undergraduate courses on operational research and related areas, and participated regularly in the teaching programme of our M.Sc. course in Statistics. In addition, he had many research students who graduated with Ph.D. degrees from our Department after having studied under his direction. Thus, in a real sense we came to regard Martin Beale as a permanent member of our Department and many of us feel he achieved as much in one day a week as might reasonably be expected of a full-time member of the staff, he was that active in both research and teaching.

Further, Sir David Cox writes:

Our Visiting Professor scheme is designed to foster contacts with research workers outside the academic sphere and particularly with industry. Martin

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entirely characteristically took his duties extremely seriously and over the years lectured regularly to final year undergraduate students of mathematics on optimization and operational research and to M.Sc. students of statistics on numerical optimization. In addition and very importantly he supervised a succession of doctoral students, 10 over the period in question. In the initial period I think students found his lectures difficult, particularly because Martin's style was not suited to the habits of British undergraduates who tend to work not so much from textbooks as from lecturers' notes which are expected to be given in a form lending themselves to coherent note-taking. An anecdote from those early years in many ways characteristic of the man is told by an ex-colleague who attended one of Martin's courses. One of the students was rather noticeably asleep. Martin at first made no comment, but towards the end of the period said to my colleague "Please wake Mr X. He will find this next bit interesting ". I feel sure this was said out of real concern and was not to be taken sarcastically. His approach to teaching at postgraduate level is recalled by E. M.

Aghedo, who was one of Martin's research students:

He was more of a father than a supervisor to me. He had the outlook of the legendary Chinese Martial Arts Teacher who would punch hard to make the pupil learn to punch and praise when he had learned. Martin Beale had the magic of making the most difficult of subjects a child's play. I was taught to be critical of any result by not being deceived by its beauty but being careful to explore fully its usefulness. He usually set me on a job like so: "Investigate the correctness of this concept. If correct, why, and if not, why not." Such was the man Martin! He was a very thorough man who believed in a high standard of work and was very uncompromising about this.

Of course he also gave much encouragement and guidance to young researchers at Scicon, as shown in a tribute from R. M. Hattersley: 'Martin taught me, with patience and kindness, most of what I know about mathematical programming. He promoted in me a lasting interest in the field, and for that I am in his debt, as no doubt are many others before me.'

The research of very many people was helped by his participation in conferences. His invited lectures were always useful but not always easy to follow, because in a single talk he would often give details of many different subjects. They were thoroughly enjoyable, however, as Martin excelled at conveying his enthusiasm, and he would not expose a topic unless he believed that it was valuable. The contents of these lectures are considered later in this memoir. When listening to papers he sat near the front of the audience, he was always attentive, and usually he asked questions. He tried much harder than most people to understand all that was said, and frequently he sought further information from speakers after their presentations. These discussions were often of more help to the speakers than to Martin, but his thirst for knowledge of mathematical programming algorithms that might be useful was unquenchable. At conferences he hardly ever missed an opportunity to listen to a relevant paper and he did not skip talks that were given by unknown researchers.

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In addition to his parallel careers at Scicon and at Imperial College, Martin contributed much to professional societies and to official committees, including the Subcommittee on Computing of the Defence Scientific Advisory Council, the Operational Research Methodology Committee and the Council of the Assessments Board, the Mathematical Sciences Subcommittee of the University Grants Committee, and three S.E.R.C. (Science and Engineering Research Council) bodies, namely the Mathematics Committee, the Operational Research Panel and the Science Board Computing Committee. It is stated by A. S. Douglas (1986) that: 'As a committee member he was always admirably to the point and never long-winded, but he did not miss much and was always willing to speak up when he felt something needed saying-a very real support for the Chairman and sometimes a useful discipline for him if he got his summing up wrong!'

He joined the Royal Statistical Society in 1950, being Honorary Secre- tary from 1970 to 1976 and a Vice-President from 1978 to 1980. He was an active member of both the Institute of Statisticians and the Operational Research Society. For many years he was Treasurer of the Statistical Dinner Club, a rather venerable institution that he served with great devotion. He contributed much to the Institute of Mathematics and its Applications, giving several invited talks at conferences, being on the editorial board of its Journal of Applied Mathematics, and serving on Council from 1980 to 1985, first as an ordinary member and then as a Vice-President. His part in the Mathematical Programming Society is described by P. Wolfe (Dantzig & Tomlin 1987):

In 1964, he was the leading spirit in organizing an 'International Sym- posium on Mathematical Programming' in London, the first such meeting held outside the United States. Subsequently, finding a small surplus of funds in their treasury, the organizers designated it the 'International Pro- gramming Fund' and chose a small international committee to hold it. This was the first step in the identification of Mathematical Programming as a professional specialty, and eventually led to the formation of the Mathe- matical Programming Society in 1972. The newly formed society recognized Martin's achievements in many ways. The first election held by the Society had to choose both its first Chairman and his successor. George Dantzig was the first Chairman and Martin the second, serving from 1974 to 1976. Prior to that, he was asked to join the board of senior editors of the Society's new journal, Mathematical Programming, a small group which included two who later became Nobel laureates. He served on the Council of the Society from 1982 to 1985 and otherwise on several of its committees. Whether he had an official role or not, he was regularly consulted on all matters of policy, and his advice was followed.

The academic excellence of his work was acknowledged by his election to the Royal Society in 1979, and by the award of the Silver Medal of the Operational Research Society in 1980. He was a member of Council of the Royal Society in 1984-85.

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PUBLISHED WORK

In his publications there are profound contributions to statistics, several techniques that have become standard procedures in mathematical programming calculations, descriptions of mathematical models for a wide range of practical applications, and explanations of the efficient use of optimization methods. Initially his writings just show remarkable skill and ingeniousness in response to real needs and academic questions, but Martin became more and more concerned that computer users generally should benefit from advances in suitable optimization algorithms. Therefore his later work includes many excellent expository conference papers, in addition to valuable new work that he continued to publish throughout his career.

Most researchers in mathematical programming propose new algorithms and study their theoretical properties, which is a far cry from the successful solution of real optimization problems, because in practice it is usual to prefer methods that are readily available through easy-to-use computer programs. Therefore the development of new methods, the

provision of general software for these methods, the use of the software for particular calculations, and the construction of the mathematical models that give the calculations are each highly important tasks. Martin was expert at all of them, but this is a story to unfold gradually. We begin with his first three papers and the research that stemmed from them, which provide a trail through his work on the algorithm development side.

His first paper, published in 1954, presents an independent discovery of the dual form of the simplex algorithm for linear programming in the following original way. One requires the least value of a linear function of n non-negative variables subject to m < n linear equality constraints. All but one of these constraints are used to eliminate (m- 1) of the variables, leaving a problem in (n-m+1) variables with one equality constraint that can be solved explicitly. If the values of the eliminated variables at this solution are non-negative then the calculation is complete. Otherwise one member of the set of eliminated variables is altered and a new iteration is begun, the change being such that each iteration increases the value of the objective function except in degenerate cases. Thus convergence is usual but reference 1955b gives an early example of cycling in linear programming calculations.

This work led to one of the first algorithms for quadratic programming (1955a, 1959c), where the calculation of the previous paragraph is generalized by allowing the objective function (F say) to be quadratic, but the constraints are as before. A linear approximation to F allows any linear programming method to provide a fruitful direction of search in the space of the variables, but now the best step along this direction may be determined by the curvature of F instead of by a constraint boundary. The key idea is to absorb this possibility into the linear programming

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framework by defining the directional derivative along the search direction to be a new variable, which gives one more variable and one more linear equality constraint. Making the new variable zero provides the optimal step that is determined by the curvature of F, and the new variable can be dropped when it is no longer useful. Holding the new variable at zero is equivalent to making future search directions conjugate to the search direction that defined the new variable, so in fact Martin proposed one of the earliest conjugate direction algorithms, but he does not mention conjugacy explicitly until his survey paper (1967 a). However, he observed in 1955 (1955 a) that the key idea is also suitable for minimizing the sum of the largest t terms from a set of s linear functions (where s and t are any positive integers with t < s), but it seems that this suggestion has not been taken further.

The use of conjugate search directions is the only technique for minimizing smooth unstructured functions of several variables that receives much attention in his publications, and it is the subject of some substantial work. Much time was given to CGAP (conjugate gradient method of approximation programming), which minimizes a general differentiable function subject to general constraints. CGAP is particularly efficient when, for any fixed values of a few of the variables, the remaining calculation is an LP (linear programming) problem. Changes to nonlinear variables are derived from linear approximations to nonlinear terms, and the constraints usually allow some of these variables to be 'independent', which means that their values are free so other variables are adjusted to satisfy the constraints. Conjugate search directions are used when the set of nonlinear independent variables is unchanged and when no derivative discontinuities arise from the LP problem that determines the linear variables; otherwise the steepest feasible descent direction is preferred. Many other details are important to efficiency (1974e, 1980b (written in 1976 with A. S. J. Batchelor)) so a later review (1978d) is recommended for a clear summary of the main ideas. Here a close relation to the reduced gradient method is explained (see also 1982b), and it is stated that 'Serious use of the current version of the system has been confined to a

sequence of oil production problems for the Kuwait Oil Company'. Later Martin seems to have decided that more successful procedures had become available for nonlinearly constrained calculations, because CGAP is not mentioned in his 1985 survey of mathematical programming systems (1985 a).

In reference 1972b he proposed the first conjugate gradient procedure for unconstrained optimization that achieves quadratic termination from a general initial search direction. The method, which is now employed in several optimization algorithms, just adds to each conjugate search direction the multiple of the initial search direction that gives orthogon- ality to the initial change in gradient. It is useful for 'restarts' that

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compensate for loss of hereditary conjugacy properties when the objective function is not quadratic, and it can provide conjugacy in constrained calculations when some earlier changes of variables have been restricted by constraint boundaries (1978e, coauthor R. Benveniste). Further work on conjugate gradients includes a device for achieving the least value of a quadratic function without any exact line searches (1985d, coauthor O. S. Brooker).

When solving nonlinear calculations Martin usually preferred to apply the 'separable programming' technique. In its basic form it requires each nonlinear function to depend on only one variable, and often this condition can be achieved by a reformulation of the calculation or by the introduction of some new variables. He found that the method is suitable for a wide range of problems (see 1965b and 1968, for example), and he increased its usefulness greatly by his work with J. A. Tomlin (1970g) on 'special ordered sets', which are as follows.

The separable programming technique makes a piecewise constant or a piecewise linear approximation to each nonlinear function of one variable. Thus each section of an approximation depends on one or two values of the underlying nonlinear function. There is a non-negative variable for every function value that is used. These variables are weights that sum to one, and the piecewise constant or piecewise linear cases are obtained by allowing only one or two adjacent variables to be non-zero. The success of this approach in practice depends on how these combinatorial conditions on the variables are achieved. It is usual to employ a 'branch and bound' method: here one solves a sequence of calculations where the constraints of each one are only linear equations and simple bounds on the variables (that can force some of the variables to take prescribed values). Usually a solution of a trial calculation does not satisfy the combinatorial conditions, and then it may be replaced by two new trial calculations whose constraints define disjoint regions that exclude the solution that has just been found but that contain the required solution. Special ordered sets provide highly successful ways of making these splits. Specifically, if only one of the m variables {v': i = 1,2, ..., m} may be non-zero, but if vi and Vj are both non-zero in a trial solution where i < j, then the '(S1) set' strategy requires v1 = v2 = ... = vk = 0 in one of the new calculations and vk+1 = vk+2 = ... =vm = 0 in the other

one, where the integer k is chosen from the interval [i,j- 1]. Alternatively, in the case of piecewise linear approximation where two adjacent variables may be non-zero, if the trial solution is infeasible because i < j-1, then the '(S2) set' strategy employs the alternatives v1 = v2 = ... = Vk_1 = 0, or vk+l = vk+2 = ... = vm = 0, where now i+1 < k <j-1.

His view of the merit of this approach changed over the years. In his book (1968), he writes: 'Separable programming is probably the most useful nonlinear programming technique', and his first paper

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on CGAP states 'Many problems can be formulated very naturally in separable terms, and I have often extolled the virtues of separable programming. It is therefore sad for me to have to admit that we have made little use of separable programming recently.' His first two papers with J. J. H. Forrest (1976a, 1978c), however, provided ideas and techniques that helped to restore separable programming to first place among his methods for nonlinear functions. They give careful consideration to the choice of the integer k of the previous paragraph and find good ways of selecting it. They propose an interpolation technique for adding more function values automatically where it is advantageous to improve the accuracy of a piecewise approximation. They introduce 'linked ordered sets' to process products of the form uf(x) directly, where u is a variable and where f(x) is a function of one variable that is treated by separable programming (see also 1980d). Their later paper (1978c), which is much easier to read than reference 1976a, gives several numerical examples to show that separable programming is suitable for calculating global solutions of non-convex optimization problems, because the approximations to nonlinear functions apply throughout their ranges. CGAP, however, employs local approximations that are usually valid only for small changes to the variables. This is the point that seems to have been decisive in Martin's eventual preference for separable programming over CGAP.

Of course the goal of this work was to improve the efficiency of actual computer programs for solving real problems, so Martin became highly expert on the practical details of techniques for integer programming, especially the branch and bound method. In an early survey article (1965 a) he writes 'So far, cutting plane methods for integer programming have been easily the most successful', but he reported enthusiastically in 1974 (1974b) that 'The most dramatic improvements during the last 3 years in the capability of general mathematical programming systems have been in the field of integer programming' and that 'Branch and bound strategies have been developed to the point where they can solve many problems completely in a moderate time.' The splitting of one continuous problem into two (branching) by the branch and bound method has been mentioned already, in order to satisfy combinatorial constraints, and the other main ingredient (bounding) is the maintenance of a bound on the final value of the objective function that allows one to rule out most of the continuous trial problems from further consideration before many branches have been made. Ideally one would not explore branches if some further work on another part of the tree would show that they cannot yield the required solution. Therefore the development of suitable practical strategies for selecting the continuous problem to solve next is crucial to efficiency. His early strategies (1965c) were unexciting, but highly successful techniques for branching and bounding are described in papers written between 1974 and 1985 (1974b, 1977a, 1979b, 1985b).

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The work on special ordered sets, for example, shows that much effort is needed to include in a mathematical programming system the ability to gain efficiency from the structure of an optimization calculation. There- fore, because most problems are now solved by general systems, it is usual to ignore structure unless there are already features in a system that can take advantage of it. However, in the days when new programs were written for new calculations and when computers were much slower, more attention was given to special structures. In particular Martin studied the 'purchase-storage problem' with G. Morton (1958): here one can buy commodities at a seasonal price, there is a known market for them in a sequence of time periods, and they can be stored for later selling but there are capacity constraints on the warehouses. It is shown that the optimal profit up to the end of a time period can be calculated easily from the optimal strategy up to the end of the previous time period, and that some parametric programming can be included too, parametric programming being the study of the effect on the solution of changes to the coefficients of the calculation. In reference 1959a he extends the 'transportation problem' to the case when each variable in the objective function is replaced by a convex function of the variable, but the proposed algorithm seems not to have been very successful. In reference 1963 a he addresses the important case when a few variables of a linear programming calculation occur in many constraints, but otherwise the problem can be partitioned into several small disjoint problems; it describes a primal algorithm that takes advantage of this structure. Another variation of the transportation problem is considered (1963 b): here there are so many sources and destinations that it is impracticable to store every source-destination distance, so instead one works with distances between relatively few key towns and from sources and destinations to their neighbouring key towns. He notes (1980e) that the special ordered set technique can be applied to a zero-one variables problem whose objective function is the ratio of two linear terms (fractional programming). After 1963, however, most of his work on structured calculations became more general, and it provided valuable contributions to the following mathematical programming systems.

Reference 1965c (coauthor R. E. Small) describes the method for integer programming that is incorporated in the C.E.I.R. LP/90/94 system. This system also solves 'decomposable' linear programming problems efficiently (1965d, coauthors P. A. B. Hughes & R. E. Small), where 'decomposable' means that, except for a few constraints that are treated specially, the problem can be separated into several small independent subproblems, so the situation is similar to the one that was studied in 1963 (1963a). Several other features of LP/90/94 are mentioned (1968). Then the UMPIRE system was developed at Scicon (1970c); this work included an independent discovery of the use of product forms and triangular decompositions of sparse basis matrices in

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linear programming (1971). Other techniques in UMPIRE for taking advantage of sparsity are special ordered sets for separable programming (1970g, 1976 a), and GUB (generalized upper bound) sets, a GUB being a bound on the sum of some of the variables subject to the condition that a variable may not be present in more than one GUB (see 1970c and 1975b, for example). UMPIRE was followed by the SCICONIC system, which is the main subject of reference 1978 a. It solves linear, separable, integer and parametric programming problems. A key feature of SCICONIC is the attention that is given to the interface between the system and the user: about 15000 lines of computer code are used to control input and output, to check some of the input, and to convert the input to a form that is suitable for the main calculation. Martin emphasized the importance of such interfaces (1970b, 1978a), because they are crucial to the work that is needed to apply a computer system to a new optimization calculation. Also he held strong views on the choice of notation for elements of matrices, and he made some specific suggestions (1974c, 1978a, 1980c) to help the formulation of complex mathematical programming problems.

Many of the invited papers that he presented at conferences describe the main techniques of these systems, and he believed that wide publicity for useful developments in mathematical programming is far more important than commercial exploitation. These talks have helped many optimization calculations generally to be solved by suitable methods. The most useful algorithms that were available 20 years ago are described clearly in his book (1968), and a sequel to it will be published soon by Wiley. He seldom included original work when he addressed a general audience, to concentrate on features that he knew from experience to be highly valuable, but at many meetings he also offered a submitted paper on new research. Thus his survey articles are rather repetitive: in particular there are many descriptions of special ordered sets for separable programming (1974b, 1976b, 1977a, 1978d, 1979b, 1980d, 1985b), and of the advantages of factored forms for the basis matrices of linear programming (1969c, 1971, 1974b, 1975b, 1976b). He always spoke enthusiastically about the techniques he believed to be best, and this point of view caused him to make the following delightfully immodest comment about his quadratic programming algorithm (1967 a): 'I am not going to make any pretence of being impartial between these methods. I will content myself with explaining how my own method works and why I think it has great advantages over all other methods.'

His publications include a wide range of examples of applications of optimization methods to investigate real problems. In many cases the Scicon computer systems were used for the actual numerical calculations, but mainly these papers consider the construction of suitable mathematical models. In 1961 he considered the accuracy of direction finders (1961a, b) the problem being to estimate the variance that occurs in

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measurements of bearings to targets. Reference 1965b (coauthors P. J. Coen & A. D. J. Flowerdew) studies the cheapest mix of raw materials that is sufficient to produce prescribed quantities of iron from four blast furnaces and a sinter plant. In reference 1966b (coauthors P. A. B. Hughes & S. R. Broadbent) the prediction of television audiences and newspaper readership from past ratings is considered to help advertisers. Reference 1970 a describes the development of a computer package for a Farm Advisory Service that is designed to give some assistance to farmers on such things as crop rotation and the scheduling of labour. A combinatorial problem with several zero-one variables is considered briefly (1972 a, coauthor J. A. Tomlin), and in 1974 there appeared a long discussion of the formulation of a military transportation problem (1974c, coauthors G. C. Beare & P. Bryan-Tatham). A model for investigating the consequences of different uses of limited resources in the national health service is described in reference 1974a (coauthors A. G. McDonald & G. C. Cuddeford), and it is mentioned again in reference 1975a. This paper, however, gives more attention to the cheapest way of expanding a pipeline network to provide greater flow capacities between prescribed nodes, which is an integer programming problem because the number of different available diameters of new pipes is small. The success of the health care model (1974 a, 1975 a) is discussed (1977b, coauthor J. Sullivan), and it is noted that a major deficiency is that there is hardly any variety in the treatment of patients who are assigned to a single class. The optimal relocation of government departments is the subject of reference 1978 b (coauthors P. S. Ayles, R. C. Blues & S. J. Wild): after the departments are grouped by using 'Communica- tions Cluster Analysis', there is a linear (or integer) programming problem to solve, except for product terms. Ways of treating these terms, including special ordered sets, are compared. He addresses the plan- ning of a network of pipes to bring natural gas ashore from wells, where the system may include compressors (at a price) to boost flows (1979c, 1983). Oil field exploration is considered (1986a) from a stochastic point of view: several statistical questions are discussed, for example the prediction of the undiscovered resources in an area by extrapolating from the number of successes and failures of drillings that have already been made.

The advantages of constructing and optimizing such models is a main theme of the 1980 Blackett Memorial Lecture (1980f). Here Martin distinguishes between routine calculations where the usefulness of numerical results has been established, and initial speculative attempts at modelling. He emphasizes that often in the latter case the main benefits of an optimization calculation are that its formulation concentrates the mind on the principal features of the field of study, and that unacceptable numerical results may yield improvements to the model. Because this point of view is tenable only if the optimization itself does not require much effort, a major purpose of this paper (and of 1969a, 1984b and

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1985 a) is to explain to operational researchers that the development of mathematical programming systems has made the optimization of the parameters of a wide range of models easy and reliable. Another interesting and general paper on modelling is reference 1980c: it gives particular attention to avoiding pitfalls that can cause severe loss of efficiency in subsequent optimization calculations. On the theoretical side of operational research he contributed a short proof that, in multi- objective assignment problems, efficient solutions cannot be dominated by linear combinations of efficient solutions (1984a).

His expert knowledge of both mathematical programming and statistics has provided some papers that do more than applying optimization techniques to statistical problems. In particular references 1962 (coauthor G. P. M. Heselden) and 1966a address Blotto games, where 'The Colonel Blotto game is a military deployment problem found in Caliban's Weekend Problems Book'. An example that he gives is the siting of missiles to defend targets against bomber attacks, when the total number of missiles is fixed and is fewer than necessary to defend all the targets adequately. Such games are combinatorial problems that can be so large that one aims only for an approximate solution. He recommends allowing the deployments to be probability distributions to achieve convexity, for this makes the calculation much easier. The main idea is illustrated by the remark that the average of 3 missiles and 5 missiles is not 4 missiles but 3 missiles half the time and 5 missiles half the time. The optimal solution of the new problem usually suggests a near optimal solution of the original calculation. Cluster analysis is the subject of reference 1969b. An algorithm for assigning points to clusters is proposed that reduces the number of clusters iteratively, and that exchanges points between clusters on each iteration until a local minimum is found. There is a discussion of statistical tests for choosing the number of clusters. Statistics is also merged with optimization in a paper on stochastic programming (1980 a, coauthors J. J. H. Forrest and C. J. Taylor). Here commodities are manufactured, stored and sold in a sequence of time periods, the only deviation from linear programming being a random element in the demand for current and future purchases that depends on previous sales (which are known). Some choices of suitable distributions for the randomness are considered, and they yield nonlinear functions that are treated by special ordered sets; consequences of these choices are illustrated by small numerical calculations. This difficult work is extended (1986b, coauthors G. B. Dantzig and R. D. Watson) to cases where some of the coefficients of the 'linear programming problem' (for example the cost of manufacturing one unit of some of the commodities in the current time period) are also stochastic.

Integer programming is combined with statistics in an efficient algorithm for discarding variables in linear regression (1967b, coauthors M. G. Kendall & D. W. Mann). Here a random variable is to be

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estimated as well as possible by a linear combination of not more than r out of p random variables, where p and r are given with r < p. This calculation would be ordinary linear regression if the set of non-zero coefficients were known, but it has to be chosen automatically to minimize the residual sum of squares. The algorithm solves a sequence of linear regression problems with various numbers of parameters, because it takes advantage of the fact that it is easy to update the solution when there is an addition to or a deletion from the set of coefficients that are allowed to be non-zero (see also 1970d and 1974f). These additions and deletions are controlled by a branch and bound strategy: for example a coefficient must be non-zero finally if, without it, the residual sum of squares is always larger than a value that has already been calculated by using at most r non-zero coefficients. Details of a general form of this integer programming procedure are given in reference 1970e. In two papers (1970f, 1982 a) he compares the algorithm in reference 1967 b with some other methods for the discarding of variables problem. Moreover, it is shown (1974d, coauthor P. C. Hutchinson) that it is futile to increase r if, at the solution, there are fewer than r non-zero coefficients.

His other contributions to statistics are mainly academic. In 1959 (1959b, coauthor C. L. Mallows) a necessary condition was found for the existence of an independent random variable Z such that X = Y+ Z, where X and Y are given random variables. Reference 1960 is a tour-de- force on confidence regions in nonlinear regression. Assuming that errors of observations are normal and unbiased it considers whether level sets of the maximum likelihood function are suitable estimates of confidence regions. Letting x be a given observation of an unknown random vector that could have the true value y, the key question is whether the probability of observing y when x is true is a good approximation to the probability of observing x when y is true. The answer depends on departures from linearity that cannot be corrected by reparameterization, and some ways of estimating the magnitude of this effect are suggested. Some approaches to the missing values problem of linear regression are studied in reference 1975c (coauthor R. J. A. Little): crudely speaking this is a least squares calculation with some unknown matrix elements. From the statistical point of view, however, these elements are observations of correlated random variables, so one can find the most likely values of coefficients even if there are no complete rows of data. It is pointed out that some algorithms for this calculation give biased estimates, and there are some illuminating numerical results. In reference 1979 a (coauthor R. G. Seeley) it is shown that algorithms for linear least squares calculations can be used to solve a problem that has serial correlation in the errors of successive observations. The 1984 Kendall Memorial Lecture (1985c) begins with a view of Sir Maurice's five careers and reminisces a little on reference 1967b. The main subject is control variables and Martingales. It is explained that when analysing

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some stochastic models it can be very helpful to make comparisons with simpler models. The idea is to express an outcome of the original model as the sum of an outcome of a simpler model, and of the difference between the two outcomes, the intention being that the first term is relatively easy to analyse and the second term has a much smaller variance than the outcome of the original model. Two examples of this technique are given: one demonstrates a way of avoiding bias and the other one is a queueing problem. As usual Martin concentrates on results that are valuable to practical calculations.

In reference 1963 a he wrote 'For the benefit of those who, like the present author, prefer numbers to formulas, I now present a miniature scale numerical example.' This inclination towards actual calculations, and this intention to communicate his work in simple terms are typical of his illustrious career. No one else has done so much to advance the successful use of mathematical programming algorithms, and his publications are a fine memorial of his achievements.

ACKNOWLEDGEMENTS

I am particularly grateful for information from Mrs E. S. L. Beale (Martin's mother), Mrs E. M. L. Beale (Martin's wife) and S. Vajda, and for permission to quote from articles by K. Bowen, G. B. Dantzig, A. S. Douglas and J. A. Tomlin. Further, I wish to thank not only the other contributors to this memoir but also the larger number of people who kindly offered material that has not been included.

The photograph reproduced was taken in 1979 by Godfrey Argent.

BIBLIOGRAPHY

1954 An alternative method for linear programming. Proc. Camb. phil. Soc. 50, 513-523. 1955a On minimizing a convex function subject to linear inequalities. Jl R. statist. Soc. B 17, 173-

184. b Cycling in the dual simplex algorithm. Nav. Res. Logist. Q. 2, 269-275.

1956 (With M. DRAZIN) Sur une note de Farquharson. C. r. Acad. Sci. Paris 243, 123-125. 1958 (With G. MORTON) Solution of a purchase-storage programme. 1. Opl Res. Q. 9, 174-187. 1959a An algorithm for solving the transportation problem when the shipping cost over each route

is convex. Nav. Res. Logist. Q. 6, 43-56. b (With C. L. MALLOWS) Scale mixing of symmetric distributions with zero means. Ann. math.

Statist. 30, 1145-1151. c On quadratic programming. Nav. Res. Logist. Q. 6, 227-243.

1960 Confidence regions in nonlinear estimation. Jl R. statist. Soc. B 22, 41-88. 1961 a Brooke variance classification system for DF bearings. J. Res. natn. Bur. Stand. D 65, 255-

261. b Estimation of variances of position lines from fixes with unknown target positions. J. Res.

natn. Bur. Stand. D 65, 263-273.

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Evelyn Martin Lansdowne Beale 43

1961 c Some uses of computers in operational research. In Proceedings of the Inaugural Meeting of the Swiss Operational Research Society, pp. 51-52. Zurich: Industrielle Organisation.

1962 (With G. P. M. HESELDEN) An approximate method of solving Blotto games. Nav. Res. Logist. Q. 9, 65-79.

1963a The simplex method using pseudo-basic variables for structured linear programming problems. In Recent advances in mathematical programming (ed. R. L. Graves & P. Wolfe), pp. 133-148. New York: McGraw-Hill.

b Two transportation problems. In Proc. Third Internat. Conf. Operational Research (ed. G. Kreweras & G. Morlat), pp. 780-788. London: English Universities Press.

1965a Survey of integer programming. J. oper. Res. Soc. 16, 219-228. b (With P. J. COEN & A. D. J. FLOWERDEW) Separable programming applied to an ore

purchasing problem. Appl. Statist. 14, 89-101. c (With R. E. SMALL) Mixed integer programming by a branch and bound technique. In Proc.

1965 IFIP Congr. (ed. W. A. Kalenich), vol. 2, pp. 450-451. Washington, D.C.: Spartan Books.

d (With P. A. B. HUGHES & R. E. SMALL) Experience in using a decomposition program. Comput. J. 8, 13-18.

1966 a Blotto games and the decomposition principle. In Theory of games techniques and applications (ed. A. Mensch), pp. 64-84. London: English Universities Press.

b (With P. A. B. HUGHES & S. R. BROADBENT) A computer assessment of media schedules. Opl Res. Q. 17, 381-411.

1967a Numerical methods. In Nonlinear programming (ed. J. Abadie), pp. 135-205. Amsterdam: North Holland.

b (With M. G. KENDALL & D. W. MANN) The discarding of variables in multivariate analysis. Biometrika 54, 357-366.

1968 Mathematical programming in practice. London: Pitmans. 1969a Mathematical programming: algorithms. In Progress in operations research (ed. J. S.

Aronofsky), vol. 3, pp. 135-173. New York: John Wiley. b Euclidean cluster analysis. In Proc. 37th Session of the ISI, Bull. Internat. Statist. Inst. 43 (2),

99-101. c Nonlinear optimization by simplex-like methods. In Optimization (ed. R. Fletcher), pp. 21-

36. London: Academic Press. 1970a A management advisory system using computerized optimization techniques. Outl. Agric. 6,

143-147. b Matrix generators and output analysers. In Proc. Princeton Symp. Mathematical Programming

(ed. H. W. Kuhn), pp. 25-36. Princeton University Press. c Advanced algorithmic features for general mathematical programming systems. In Integer and

nonlinear programming (ed. J. Abadie), pp. 119-137. Amsterdam: North Holland. d Computational methods for least squares. In Integer and nonlinear programming (ed. J.

Abadie), pp. 213-227. Amsterdam: North Holland. e Selecting an optimum subset. In Integer and nonlinear programming (ed. J. Abadie), pp.

451-462. Amsterdam: North Holland. f Note on procedures for variable selection in multiple regression. Technometrics 12, 909-

914. g (With J. A. TOMLIN) Special facilities in a general mathematical programming system for

non-convex problems using ordered sets of variables. In Proc. Fifth Internat. Conf. Operational Research (ed. J. Lawrence), pp. 447-454. London: Tavistock Publications.

h (Editor) Applications of mathematical programming techniques. New York: Elsevier. 1971 Sparseness in linear programming. In Large sparse sets of linear equations (ed. J. K. Reid), pp.

1-15. London: Academic Press. 1972 a (With J. A. TOMLIN) An integer programming approach to a class of combinatorial problems.

Math. Prog. 3, 339-344. b A derivation of conjugate gradients. In Numerical methods for nonlinear optimization (ed. F.

A. Lootsma), pp. 39-43. London: Academic Press. 1974 a (With A. G. MCDONALD & G. C. CUDDEFORD) Balance of care: some mathematical models of

the National Health Service. Br. med. Bull. 30, 262-271. b The significance of recent developments in mathematical programming systems. In

Mathematical programming in theory and practice (ed. P. L. Hammer & G. Zoutendijk), pp. 11-29. Amsterdam: North Holland.


1974c (With G. C. BEARE & P. BRYAN-TATHAM) The DOAE reinforcement and redeployment study: a case study in mathematical programming. In Mathematical programming in theory and practice (ed. P. L. Hammer & G. Zoutendijk), pp. 417-442. Amsterdam: North Holland.

d (With P. C. HUTCHINSON) Note on constrained optimum regression. Appl. Statist. 23, 208- 210.

e A conjugate gradient method of approximation programming. In Optimization methods for resource allocation (ed. R. Cottle & J. Krarup), pp. 261-277. London: English Universities Press.

f The scope of Jordan elimination in statistical computing. I.M.A. Bull. 10, 138-140. 1975 a Some uses of mathematical programming systems to solve problems that are not linear. Opl

Res. Q. 26, 609-618. b The current algorithmic scope of mathematical programming systems. Math. Prog. Stud. 4,

1-11. c (With R. J. A. LITTLE) Missing values in multivariate analysis. Jl R. statist. Soc. B 37,129-145.

1976a (With J. J. H. FORREST) Global optimization using special ordered sets. Math. Prog. 10, 52-69.

b Optimization techniques based on linear programming. In Optimization in action (ed. L. C. W. Dixon), pp. 447-466. London: Academic Press.

1977 a Integer programming. In The state of the art in numerical analysis (ed. D. A. H. Jacobs), pp. 409-448. London: Academic Press.

b (With J. SULLIVAN) Mathematical and computational concepts of a macro-economic model of the balance of health care. In Systems science in health care (ed. A. M. Coblentz & J. R. Walter), pp. 313-319. London: Taylor and Francis.

1978a Mathematical programming systems. In Numerical software-needs and availability (ed. D. A. H. Jacobs), pp. 363-376. London: Academic Press.

b (With P. S. AYLES, R. C. BLUES & S. J. WILD) Mathematical models for the location of government. Math. Prog. Stud. 9, 59-74.

c (With J. J. H. FORREST) Global optimization as an extension of integer programming. In Towards global optimization, II (ed. L. C. W. Dixon & G. P. Szego), pp. 131-149. Amsterdam: North Holland.

d Nonlinear programming using a general mathematical programming system. In Design and implementation of optimization software (ed. H. J. Greenberg), pp. 259-279. Alphen aan den Rijn: Sijthoff & Noordhoff.

e (With R. BENVENISTE) Quadratic programming. In Design and implementation of optimization software (ed. H. J. Greenberg), pp. 249-258. Alphen aan den Rijn: Sijthoff & Noordhoff.

1979 a (With R. G. SEELEY) First order auto-regressive regression analysis. In Forecasting (ed. 0. D. Anderson), pp. 167-175. Amsterdam: North Holland.

b Branch and bound methods for mathematical programming systems. In Discrete optimization, II (ed. P. L. Hammer, E. L. Johnson & B. H. Korte). Amsterdam: North Holland. (Ann. Discrete Math. 5, 201-219.)

c The optimisation of gas gathering pipeline networks. I.M.A. Bull. 15, 126-128. 1980a (With J. J. H. FORREST & C. J. TAYLOR) Multi-time-period stochastic programming. In

Stochastic programming (ed. M. A. H. Dempster), pp. 387-402. London: Academic Press.

b (With A. S. J. BATCHELOR) A revised method of conjugate gradient approximation programming. In Survey of mathematical programming (ed. A. Prekopa), pp. 329-346. Budapest: Akademie Kiado.

c Pitfalls in optimization methods. In Pitfalls of analysis (ed. G. Majone & E. S. Quade), pp. 70-88. Chichester: John Wiley.

d Branch and bound methods for numerical optimization of non-convex functions. In COMPSTAT 1980 (Proc. Fourth Symp. Comp. Stat.) (ed. M. M. Barritt & D. Wishart), pp. 11-20. Vienna: Physica Verlag.

e Fractional programming with zero-one variables. In Extremal methods and systems analysis, lecture notes in economics and mathematical systems, 174 (ed. A. V. Fiacco & K. 0. Kortanek), pp. 430-432. Berlin: Springer-Verlag.

f The Blackett Memorial Lecture 1980. Operational research and computers: a personal view. J. oper. Res. Soc. 31, 761-767.

1982 a Elimination of variables. In Encyclopedia of statistical sciences, vol. 2, pp. 482-485. New York: John Wiley.

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Evelyn Martin Lansdowne Beale 45

1982b Algorithms for very large nonlinear optimization problems. In Nonlinear optimization 1981 (ed. M. J. D. Powell), pp. 281-292. London: Academic Press.

1983 A mathematical programming model for the long-term development of an off-shore gas field. Discrete Appl. Math. 5, 1-9.

1984 a Note on 'A special multi-objective assignment problem' by D. J. White. J. oper. Res. Soc. 35, 769-770.

b Mathematical programming. In Developments in operational research (ed. R. W. Eglese & G. K. Rand), pp. 1-10. Oxford: Pergamon Press.

1985a The evolution of mathematical programming systems. J. oper. Res. Soc. 36, 357--366. b Integer programming. In Computational mathematical programming (ed. K. Schittkowski),

pp. 1-24. Berlin: Springer-Verlag. c The Kendall Memorial Lecture. Regression: a bridge between analysis and simultation.

Statistician 34, 141-154. d (With O. S. BROOKER) The use of hypothetical points in numerical optimization. Math. Prog.

Stud. 25, 28-45. 1986a Optimization methods in oil and gas exploration. I.M.A. J. appl. Math. 36, 1-10.

b (With G. B. DANTZIG & R. D. WATSON) A first order approach to a class of multi-time period stochastic programming problems. Math. Prog. Stud. 27, 103-117.

REFERENCES TO OTHER AUTHORS

Bowen, K. 1986 Professor E. M. L. Beale-a personal tribute. O.R. Newsletter (February), pp. 8-9. Dantzig, G. B. & Tomlin, J. A. 1987 (ed.) E. M. L. Beale, FRS; friend and colleague. Math. Prog.

38. 117-131. Douglas, A. S. 1986 Obituary. Professor Evelyn Martin Lansdowne Beale, FRS, FIMA. I.M.A.

Bull. 22, 120-122. Obituary: Evelyn Stewart Lansdowne Beale. The Times, 26 January 1972.

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