$ in red. a) Initial trivial tour given by $ \sigma = (1, \dots, 10) $. b–d) Intermediate solutions. e) Convergence to an orthogonal matrix which is “close” to a permutation matrix with respect to any matrix norm. f) Extraction of the corresponding permutation matrix. The initial tour was transformed into the optimal tour by the gradient flow.[]{data-label="fig:P flow"}](P2){width="80.00000%"}

![Traveling salesman problem with 10 cities solved using the gradient flow . The original positions of the cities are shown in black, the positions transformed by the orthogonal matrix $ P $ in red. a) Initial trivial tour given by $ \sigma = (1, \dots, 10) $. b–d) Intermediate solutions. e) Convergence to an orthogonal matrix which is “close” to a permutation matrix with respect to any matrix norm. f) Extraction of the corresponding permutation matrix. The initial tour was transformed into the optimal tour by the gradient flow.[]{data-label="fig:P flow"}](P3){width="80.00000%"}

\

![Traveling salesman problem with 10 cities solved using the gradient flow . The original positions of the cities are shown in black, the positions transformed by the orthogonal matrix $ P $ in red. a) Initial trivial tour given by $ \sigma = (1, \dots, 10) $. b–d) Intermediate solutions. 