Presentation

My name is Hubert Villuendas and I am a PhD student in applied mathematics at the Jean Kuntzmann Laboratory and the Grenoble Computer Science Laboratory, under the supervision of Jérôme Malick and Mathieu Besançon.

My thesis focuses on methods and applications of mixed-integer non-linear optimisation. More specifically, I focus on mixed-integer conic optimisation, where decision variables take integer values and constraints can be expressed as belonging to a convex cone \(\mathcal{K}\subseteq\mathbb{R}^m\):

where \(A\in\mathbb{R}^{n\times m}\) and \(b\in\mathbb{R}^m\).

Conic optimisation generalises linear optimisation by introducing structured non-linearities that capture a large number of essential problems in operational research, combinatorial optimisation, and data science.

My most recent research focuses on Distributionally Robust Optimization for data-driven constrained problems. Consider a mathematical program over a constrained feasible set \(\mathcal{X}\) with a non-linear stochastic objective \(\mathbb{E}_{\xi\sim\mathbb{P}}[f(x,\xi)]\), where the loss function \(f\) depends on an unknown random variable, a scenario \(\xi\in\Xi\subseteq\mathbb{R}^d\). Given training data \(\widehat{\xi}_1,\dots,\widehat{\xi}_N\), we build the empirical distribution \(\widehat{\mathbb{P}}_N=(\delta_{\widehat{\xi}_1}+\dots+\delta_{\widehat{\xi}_N})/N\) and we aim to find a feasible \(x\in\mathcal{X}\) that minimizes the objective under the worst possible distribution within a neighbourhood of the empirical distribution: