### Pierre Hansen

As with other researchers, Pierre Hansen is just as fascinated now by a problem raised in his “Agrégation” thesis at the University of Brussels as he was in 1974, that is locating and establishing the optimal number of warehouses in a distribution network. Nowadays, however, the complexity of the problem has increased exponentially. In 1974, the challenge was to determine optimal locations, using exact methods, from among some 100 potential spots that would serve 1,500 user groups. Now, Pierre Hansen can optimize the choice, still using exact methods, from among 7,000 potential spots to serve 7,000 user groups. This translates into approximately 49 million variables and just as many constraints to be considered. In terms of size, this is a record.

As regards heuristic methods, problems four as large (15,000 potential locations) can now be solved. Moreover, guarantees as to the optimal quality of the solution are now provided with a margin of error of 0.4 percent. In a nutshell, the exact method consists in finding the optimal solution along with proof that it is optimal, which is a hard task if there are many solutions locally optimal. Solving such problems is the aim of g*lobal optimization *(in discrete or continuous variables). “It’s like asserting that Mont Blanc is the highest summit in Europe and also having proof in hand,” says Professor Hansen. In contrast, the heuristic method does not provide proof of the solution.

According to Pierre Hansen, this spectacular progress can be explained by the development of powerful IT tools as well as by the relentless work of researchers in finding new technical and mathematical methods (i.e., algorithms) to solve problems. Hansen adds that the strength and beauty of mathematics largely lie in the countless applications it offers in all areas of human activity as well as in other branches of mathematics. When someone does high-power math in an operations research centre like GERAD, problems and applications, be they theoretical or practical, are drawn like iron filings to a magnet.

His own research has applications in at least fifteen disciplines. These include electricity, acoustics, geometry, location, pure economics, biology, graph theory, probability, artificial intelligence, mathematical chemistry, chemistry of solids, and marketing. The applications concern a wide variety of problems such as the mooring of ships in the port of Hong Kong, the resolution of a geometry problem open since 1950 (discovery with his colleague Charles Audet of the octagon with smallest diameter and unit sides, improving on “Vincze’s wife’s octagon”), the optimal dimensions of a listening room, or the classification of clients of a shopping mall.

The work conducted in this latter area is a *data mining *application. Data mining is like panning for gold nuggets in quicksand. Basically, to extract useful and profitable information from masses of data that do not appear to make any sense and or have any use, a series of methods of classification and discrimination devised by Hansen during his long career can be applied. The Bicriterion Cluster Analysis method leads to the discovery of classes of entities, called clusters, that are homogeneous and well separated within a set of objects. It applies to classification of psychological tests, shares or geometric figures.

A continual and healthy communication is maintained between mathematical methods and applications, according to Pierre Hansen. “We solve theoretical problems, publish the solutions in major journals and then we apply them to other theoretical or practical problems.” For example, an algorithm for Non-convex Quadratic Programming was first published in the prestigious journal*Mathematical Programming *in January 2000. Applications were soon to follow, in the oil industry, in finance, in supply chain management, in data mining and in geometry, leading to further papers.

A major source of satisfaction for Pierre Hansen is the meta-heuristic method known as Variable Neighbourhood Search, developed since 1997 together with Nenad Mladenović of the Mathematical Institute of Serbian Academy of Science in Belgrade. The method provides a general framework for doing heuristics in which it is easy to understand what is happening and where there are few or no parameters. It is already used in a variety of applications and has been cited more than 80 times in academic journals.

Variable Neighbourhood Search method served, for instance, to develop the AutoGraphiX system, which was the subject of the doctoral thesis of Hansen’s student Gilles Caporossi. AutoGraphiX is a method of computer-assisted discovery of conjectures in graph theory.

“It’s just like Archimedes”, observes Professor Hansen. “You have to know what you want to prove, that is the conjecture. Then you prove it. Basically, you prove something that is probably true.” It is interesting to note that the external examiner on Gilles Caporossi’s thesis was Herbert A. Simon, who won the Nobel Prize for Economics in 1978. What’s more, this method has already led to several hundred conjectures, many of witch have been proven.

Pierre Hansen is one of the strongest defenders of the GERAD operations research centre that he headed for five years. “Operations research is not present in all universities,” he notes. “Renowned researchers often feel isolated, even in major universities. Montreal is the city that counts the largest number of operations researchers in the world. GERAD is among the five most important operations research centers in the world in terms of scientific output. That means that when we have a specific problem, we can often find the world authority on the issue in the very next office. That is an invaluable asset!”

*Extract from an article published in GERAD Newsletter, volume 1, number 1, February 2004.*

Since 2009, research conducted through the Chair has led to the publication of 90 articles in international scientific journals; another 3 were accepted for publication and 13 more were submitted. In addition, 2 books were edited and 12 book chapters as well as refereed proceedings from 1 conference were published. In the same period, chairholder Pierre Hansen presented 16 plenary, semi-plenary or wrap-up conferences at international conventions or symposiums. The Chair collaborates with 81 foreign researchers (from 18 countries) and 17 Canadian researchers, not counting students. Chair members also supervised 1 postdoctoral student and oversaw 7 doctoral dissertations and 3 master’s theses during that timeframe. Two doctoral dissertations and one master’s thesis are currently underway.

Fifteen international interns completed their projects within the framework of the Chair. Pierre Hansen organized or presided over 7 international symposiums, conventions or workshops. He is currently editor or associate editor of 15 scientific journals. Pierre Hansen and his team also won several scientific awards since 2009, including HEC Montréal’s Pierre Laurin Award in 2010 (Pierre Hansen, ex aequo with Gilbert Laporte) and again in 2013 (Pierre Hansen, for the excellence of his research over his entire career). It should be noted that the number of publications from Chair members or their students that were either published, submitted or accepted—which was already high for the 2004-2009 timeframe (130)—has increased significantly for the 2009-2015 period (142).

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