My research sits at the intersection of convex and combinatorial optimization, with applications to robotics, motion planning, and optimal control. Specifically, I study optimal decision making in circumstances where discrete and continuous choices have to be made simultaneously. I work on these problems on a mathematical and numerical level: I develop modeling frameworks, transcriptions as numerical optimizations, and solution algorithms. The questions at the core of my research are also central in machine learning and AI: foundation and large-language models have recently unlocked unprecedented opportunities for providing our robots with common sense and long-term reasoning, but have also highlighted the lack of optimization methods that can reliably and automatically generate large amounts of high-quality training data.

The main outcome of my PhD has been Graphs of Convex Sets (GCS): a modelling and decision-making framework that efficiently combines graph search and convex optimization. Formally, a GCS is a graph where each vertex is paired with a convex program, and each edge couples two programs through additional convex costs and constraints. Any optimization problem over an ordinary weighted graph can be extended to GCS in a natural way, yielding a new class of problems with a wide range of applications. My main contribution has been a general methodology to formulate any GCS problem as a lightweight mixed-integer program with tight convex relaxation.

The shortest-path problem in GCS is especially important in control and robotics, since it encompasses as special cases many trajectory-optimization and motion-planning problems. Through a single convex program, we can now design globally optimal trajectories for a car traversing a maze in minimum time, or solve intricate bi-manual manipulation problems.