**Adrien Taylor’s UCLouvain PhD thesis on optimisation, a branch of applied mathematics, earned two research awards, including the FNRS-IBM Innovation Award. **

Optimisation is about finding the best way – depending on your criteria – to reach or at least get as close as possible to a goal. ‘Imagine your goal is to get to the office in the most satisfactory way’, says Adrien Taylor, a UCLouvain researcher in applied mathematics. ‘To do this, you have several options and thus parameters to evaluate: transport mode (car, bike, public, etc.), route, schedule, etc. Your goal may be to find the quickest, most economical or least polluting route. This notion, "the most" or "the least something", is optimisation.’

#### Finding the best solutions

Mathematical optimisation affects many fields. ‘Many computer or scientific problems can be seen as optimisation problems’, Dr Taylor continues. ‘Examples are rocket trajectory, internet search engine design, software that converts measurements into medical imagery, and so on. Often, it’s not really possible to find the perfect solution to such complex problems. The challenge is to get as close as possible.’

To do so, engineers and mathematicians use algorithms. An algorithm is a descriptive process, or instructions for using calculations. It’s a bit like a recipe that explains how and in what order to use ingredients (calculations) to arrive at a result. Now there are entire libraries of algorithms. How can they be rigorously compared? This was the central question of Dr Taylor's thesis.

#### Generating worst-case scenarios

‘We developed a methodology that allows for a systematic algorithm analysis by a computer’, he explains. ‘By applying this methodology, the computer generates worst-case scenarios. Which means that for a given family of optimisation problems – which the algorithm is supposed to solve – our methodology generates the problems on which this algorithm will work the least. So, if the computer generates only the worst cases on which your algorithm works satisfactorily, it means that it will always work satisfactorily (on this class of problems).’

The study of worst cases isn’t new. However, these analyses are rarely completely representative of reality. They can, for example, give an extremely pessimistic view of what’s happening in practice. ‘Our methodology generates worst cases but does so perfectly. That is, the results obtained can no longer be improved. Because, in a way, our methodology adopts the most optimistic point of view possible among the pessimistic points of view.’

In other words, the methodology doesn’t waste time with algorithms that won’t possibly help you. This advance appealed to the juries of the FNRS-IBM Innovation and ICTEAM awards,2 both of which Dr Taylor recently received. ‘I was very lucky with the environment and the people I worked with during my thesis’, he says. ‘My supervisors at UCLouvain, particularly Profs François Glineur and Julien Hendrickx, have provided invaluable support.’

#### Back to applications?

Although he focuses on very complex mathematics, Dr Taylor reiterates that he’s an engineer, not a mathematician, by training. ‘For me, math is more of a means than an end in itself. At the beginning of my PhD, I thought I’d work on concrete applications. And there were some. One of the algorithms we developed (thanks to our methodology) is used in medical imaging (MRI and CT) by other teams. But in the end I focused on the analysis and evaluation of the method rather than actual applications.’

He’s currently working as a researcher as part of SIERRA,3 a French machine learning laboratory that focuses on optimisation. ‘I'm more into theorising, designing and analysing algorithms’, he says. ‘But after my postdoctorate, I don’t rule out a return to applications, depending on the opportunities that arise.’

(1)‘Convex Interpolation and Performance Estimation of First-order Methods for Convex Optimization’, thesis defended 11 January 2017 in Louvain-la-Neuve. (2)Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) is part of UCLouvain. (3)SIERRA is a joint team of the École Normale Supérieure de Paris, the French National Centre for Scientific Research (CNRS), and the French Institute for Research in Computer Science and Automation (Inria).

## A glance at Adrien Taylor's bio

2011: Dual Master’s Degree in Engineering and Applied Mathematics, UCLouvain and KU Leuven

2012: R&D Advisor, Secura

2016: PhD in Applied Mathematics, UCLouvain

2017: Researcher, UCLouvain

Since 2017: Researcher, École Normale Supérieure and Inria (Paris)

2018: Finalist, Tucker Prize for PhD thesis

2018: Winner, ICTEAM Award for PhD thesis

2018: Winner, FNRS-IBM Innovation Award