long and rich history in the areas of computer science, optimization theory, and computational complexity, and has received decades of interest [@Cit:cook]. This combinatorial optimization problem arises in a wide variety of applications related to genome map construction, telescope management, and drilling circuit boards. The TSP also naturally occurs in applications related to target tracking [@Cit:target_tracking], vehicle routing, and communication networks to name a few. We refer the reader to [@Cit:cook; @Cit:TSP_sahai] for further details.

In its basic form, the statement of the TSP is exceedingly simple. The task is to find the shortest Hamiltonian circuit through a list of cities, given their pairwise distances. Despite its simplistic appearance, the underlying problem is NP-hard [@Cit:karp]. Several heuristics have been developed over the years to solve the problem [@Cit:cook] including ant colony optimization, cutting plane methods, Christofides heuristic algorithm, and the Lin–Kernighan heuristic.

In [@Cit:TSP_sahai], inspired by dynamical systems theory, the authors construct novel orthogonal relaxation based approximations to the TSP. In particular, the constructed dynamical system captures the flow on the manifold of orthogonal matrices and ideally converges to a permutation matrix that minimizes the tour length. However, in general, the flow typically converges to local minima that are not 