🤖 AI Summary
For large-scale chain-structured optimization problems—such as robot localization—existing semidefinite programming (SDP) relaxation methods suffer from O(n³) computational complexity, hindering real-time applicability. This paper introduces the first SDP solver that integrates chordal decomposition with the alternating direction method of multipliers (ADMM), leveraging the inherent chordal sparsity of such problems to construct a distributed SDP framework with linear-time complexity. We provide theoretical guarantees that the proposed method preserves global optimality while reducing time complexity to O(n). Experiments on simulated localization tasks demonstrate that our algorithm achieves speedups of over an order of magnitude compared to standard SDP solvers, exhibits robustness to poor initializations, and reliably converges to the globally optimal solution. This work establishes the first SDP-based paradigm for large-scale nonconvex estimation that simultaneously achieves linear computational complexity and rigorous theoretical optimality guarantees.
📝 Abstract
In recent years, many estimation problems in robotics have been shown to be solvable to global optimality using their semidefinite relaxations. However, the runtime complexity of off-the-shelf semidefinite programming (SDP) solvers is up to cubic in problem size, which inhibits real-time solutions of problems involving large state dimensions. We show that for a large class of problems, namely those with chordal sparsity, we can reduce the complexity of these solvers to linear in problem size. In particular, we show how to replace the large positive-semidefinite variable with a number of smaller interconnected ones using the well-known chordal decomposition. This formulation also allows for the straightforward application of the alternating direction method of multipliers (ADMM), which can exploit parallelism for increased scalability. We show for two example problems in simulation that the chordal solvers provide a significant speed-up over standard SDP solvers, and that global optimality is crucial in the absence of good initializations.