Convex relaxations for large-scale graphically structured nonconvex problems with spherical constraints: An optimal transport approach

In this paper we derive a moment relaxation for large-scale nonsmooth optimization problems with graphical structure and spherical constraints. In contrast to classical moment relaxations for global polynomial optimization that suffer from the curse of dimensionality we exploit the partially separable structure of the optimization problem to reduce the dimensionality of the search space. Leveraging optimal transport and Kantorovich--Rubinstein duality we decouple the problem and derive a tractable dual subspace approximation of the infinite-dimensional problem using spherical harmonics. This allows us to tackle possibly nonpolynomial optimization problems with spherical constraints and geodesic coupling terms. We show that the duality gap vanishes in the limit by proving that a Lipschitz continuous dual multiplier on a unit sphere can be approximated as closely as desired in terms of a Lipschitz continuous polynomial. The formulation is applied to sphere-valued imaging problems with total variation regularization and graph-based simultaneous localization and mapping (SLAM). In imaging tasks our approach achieves small duality gaps for a moderate degree. In graph-based SLAM our approach often finds solutions which after refinement with a local method are near the ground truth solution.