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Pose Graph Optimization (PGO) is the problem of estimating a set of posesfrom pairwise relative measurements. PGO is a nonconvex problem, and currentlyno known technique can guarantee the computation of an optimal solution. Inthis paper, we show that Lagrangian duality allows computing a globally optimalsolution, under certain conditions that are satisfied in many practical cases.Our first contribution is to frame the PGO problem in the complex domain. Thismakes analysis easier and allows drawing connections with the recent literatureon unit gain graphs. Exploiting this connection we prove non-trival resultsabout the spectrum of the matrix underlying the problem. The secondcontribution is to formulate and analyze the dual problem in the complexdomain. Our analysis shows that the duality gap is connected to the number ofeigenvalues of the penalized pose graph matrix, which arises from the solutionof the dual. We prove that if this matrix has a single eigenvalue in zero, then(i) the duality gap is zero, (ii) the primal PGO problem has a unique solution,and (iii) the primal solution can be computed by scaling an eigenvector of thepenalized pose graph matrix. The third contribution is algorithmic: we exploitthe dual problem and propose an algorithm that computes a guaranteed optimalsolution for PGO when the penalized pose graph matrix satisfies the Single ZeroEigenvalue Property (SZEP). We also propose a variant that deals with the casein which the SZEP is not satisfied. The fourth contribution is a numericalanalysis. Empirical evidence shows that in the vast majority of cases (100% ofthe tests under noise regimes of practical robotics applications) the penalizedpose graph matrix does satisfy the SZEP, hence our approach allows computingthe global optimal solution. Finally, we report simple counterexamples in whichthe duality gap is nonzero, and discuss open problems.

Pose Graph Optimization in the Complex Domain: Lagrangian Duality, Conditions For Zero Duality Gap, and Optimal Solutions