A Unified and Scalable Method for Optimization over Graphs of Convex Sets

Abstract: A Graph of Convex Sets (GCS) is a graph in which vertices are associated with convex programs and edges couple pairs of programs through additional convex costs and constraints. Any optimization problem over an ordinary weighted graph (e.g., the shortest-path, the traveling-salesman, and the minimum-spanning-tree problems) can be naturally generalized to a GCS, yielding a new class of problems at the interface of combinatorial and convex optimization with numerous applications. In this paper, we introduce a unified method for solving any such problem. Starting from an integer linear program that models an optimization problem over a weighted graph, our method automatically produces an efficient mixed-integer convex formulation of the corresponding GCS problem. This formulation is based on homogenization (perspective) transformations, and the resulting program is solved to global optimality using off-the-shelf branch-and-bound solvers. We implement this framework in GCSOPT, an open-source and easy-to-use Python library designed for fast prototyping. We illustrate the versatility and scalability of our approach through multiple numerical examples and comparisons.