András Prékopa, mathematician and operations researcher of extraordinary talent and energy, passed away on September 18, 2016, at the age of 87. We lost in him a brilliant researcher, a devoted mentor, and a great friend.
András Prékopa was born on September 11, 1929, in Nyiregyháza, Hungary. He obtained the Master of Sciences degree in mathematics, physics, and descriptive geometry in 1952 from the University of Debrecen. In 1956, at the Institute for Applied Mathematics of the Hungarian Academy of Sciences, he defended his Ph. D. dissertation, under the supervision of Alfréd Rényi. In 1971, he obtained his higher doctorate degree. From 1956 to 1968, he was a professor of Eötvös Loránd University and subsequently became a professor of the Budapest Technical University. In addition to the university responsibilities, from 1970 to 1985, he was the head of the Computer Science Center of the Hungarian Academy of Sciences, and subsequently the founder and head of the Applied Mathematics Division of the Computer Science and Automation Research Department at the Eötvös Loránd University and became its first chairman. From 1985 to 2015, he was a Distinguished Professor of operations research and statistics at the Rutgers Center of Operations Research, Rutgers University in New Brunswick, New Yersey. Until his retirement in 2015, he was also affiliated with the Department of Management Science and Information Systems.
András Prékopa was member of the Hungarian Academy of Sciences, a foreign member of the National Academy of Engineering of Mexico, a fellow of the Econometric Society, a member of the International Statistical Institute, and the honorary president of the János Bolyai Mathematical Society and the Hungarian Operations Research Society. He was the founder of the sequence of international conferences on stochastic programming (held every three years since the 1981 meeting in Kőszeg, Hungary). He was also a co-founder and chair (1981-1989) of the Committee of Stochastic Programming within the Mathematical Optimization Society.
András has published more than 350 papers and 15 books. His results, starting with his first paper, published while he was an undergraduate student in 1950, have ranged over several areas of mathematics and operations research. Below is a brief summary of his most influental contributions.
As a graduate student, András worked on Poisson processes, their generalizations, random set functions, and stochastic integrals. In his Ph. D. dissertation, as one of the first researchers he developed the theory of random measures and random set functions. He defined the notion of a random set function of independent increments in a general abstract space. He obtained stochastic counterparts of the measure extension theorems, introduced the notion of a characteristic functional, proved Radon-Nikodym type theorems, and defined stochastic Lebesgue integrals with random measures and deterministic, measurable integrands. He proved several deep results on random point distributions of Poisson type in abstract spaces. He also initiated the theory of marked Poisson processes. At that time, Hungarian sciencists were not allowed to publish abroad and his papers appeared in local journals.
In the 1960s András became one of the initiators of stochastic programming and one of its main contributors. When Charnes and Cooper first formulated chance constraints, they imposed them individually on each stochastic constraint, neglecting stochastic dependence among the random variables. Prékopa's general formulation of joint chance constraints took the dependence into account. One of his main results concerns convexity theory of stochastic programming problems with joint chance constraints. He introduced the concept of logarithmic concave measures and proved that if a probability measure is generated by a log-concave probability density function, then the measure is log-concave (an inequality that is key for this result is known as the Prékopa-Leindler inequality). These breakthrough results let to the proof of convexity of a wide class of stochastic programming problems with probabilistic constraints. For problems with chance constraints involving discrete distributions, he introduced the concept of a p-efficient point and successfully used it to develop effective methods for solving such problems. Nowadays, almost every work on problems with chance constraints uses, in one way or another, the ideas of András Prékopa.
In the mid-1980-s András invented new ways to obtain sharp bounds on the probability of the union and other Boolean functions of random events. He discovered that bounds using few terms from the inclusion-exclusion formula are optimal values of linear programming problems, which are moment problems of a certain type. Then he extended the analysis of moment problems by using linear programming theory, both in the univariate and multivariate cases. In the univariate case he fully described the structure of the dual feasible bases for three cases: the probability of the union, and the probability that at least r or exactly r out of n events occur. The results allowed for closed-form solution of small scale problems, as well as the development of efficient dual type algorithms for large problems. Linear programming turned out to be the right tool to obtain best bounds. The bounds can be used for approximate solution of stochastic programmming problems with probabilistic constraints.
András succesfully combined his theoretical work with various applications. He often expressed his strong conviction that theory and applications have to come together motivating and reinforcing each other. At the beginning of the 1960s he worked out an original inventory control model using order statistics. The model became widely used, with major economic impact in Hungary. He created novel water reservoir system design models based on stochastic programming. He applied stochastic network theory to power systems. He also worked out a daily scheduling model for electricity production in an interconnected system with thermal power plants. He used moment bounds for analyzing the reliability of telecommunication and transportation networks.
András was very much interested in the history of science and used all opportunities to share his knowledge with others. He wrote about the life and works of Gyula Farkas, the eminent Hungarian mathematician and physicist of the 19th and 20th centuries, who developed the theory of linear inequalities and published fundamental papers on the mechanical equilibrium. In connection with that, András wrote a paper on the origins of nonlinear optimization. In a volume devoted to the memory of János Bolyai, Prékopa gave an account of the discovery of non-Euclidean geometry in the first half of the 19th century, which changed the course of the development of mathematics and had an impact on the history of human culture. In further papers, he discusses the relationship between mathematics and the history of culture.
András supervised a record number of doctoral students: 52. His advisees are university professors and researchers in many countries on three continents. Among his books, most influental are those based on his undergraduate courses. His introductory book on linear programming was published in 1968, ten years after András gave his first course on the subject at the University of Budapest. It presents linear programming in a mathematically exact, elegant, and didactic way and is still in use in Hungary. His 1995 monograph stochastic programming is a high-level, very informative book with comprehensive coverage of the area that has become a standard reference and a text for doctoral courses.
The talents and achievements of András Prékopa were recognized early on by the
scientific community. He received the Grünwald Géza prize of the Bolyai
Mathematical Society for his Ph. D. dissertation. He was the recipient of the
INFORMS President's Award (2014), the Khachiyan Prize (2012), the EURO Gold Medal
(2003), as well as many state and society distinctions and prizes. But the highest
prize, one that never fades away, is the wide influence of his research ideas.
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Andrzej Ruszczynski, Department of Management Science and Information Systems, Rockafeller Road, Livingston Campus, Rutgers University, Piscataway, NJ 08854, USA, firstname.lastname@example.org