competitive when compared to state-of-the-art heuristics. Inspired by this continuous relaxation, the authors compute the solution to a two-sided orthogonal Procrustes problem [@Cit:Procrustes-book] that relaxes the TSP to the manifold of orthogonal matrices. They then combine the Procrustes approach with the Lin–Kernighan heuristic [@Cit:LKH] for computing solutions of the TSP. Additionally, the authors use set-oriented methods to study the stability of optimal solutions and their stable manifolds, thereby providing insight into the associated basins of attraction and the resulting computational complexity of the problem.

Given a list of $ n $ cities $ \{ C_{1}, C_{2}, \dots, C_{n} \} $ and the associated distances between cities $ C_i $ and $ C_j $, denoted by $ d_{ij} $, the TSP aims to find an ordering $ \sigma $ of $ \{ 1, 2, \dots, n \} $ such that the tour cost, given by $$\label{eq:basic_cost}
    c = \sum_{i=1}^{n-1} d_{\sigma(i), \sigma(i+1)} + d_{\sigma(n), \sigma(1)},$$ is minimized. For the Euclidean TSP, for instance, $ d_{ij} = ||x_i - x_j||_2 $, where $ x_i \in \R^d $ is the position of $ C_i $. In general, however, the distance matrix $ D = (d_{ij}) $ does not have to be symmetric (for example 