在旅行商问题 (TSP) 中为 scipy.optimize.linprog Python function 定义线性规划约束

Question

I am trying to solve the Linear Programming formulation of the Traveling Salesman Problem (TSP) using scipy.optimize.linprog in Python .

This document clearly defines the problem and, though I understand the idea, I can't figure out how to translate it into the required parameters.

I've already flattened the distance matrix, so the function should be fine. What about the constraint matrix and constraint vector?

I've tried this...

n = len(d)

TSP_c = [d[i][j] for i in range(n) for j in range(n)]

E = [(i,j) for i in range(len(d)) for j in range(len(d))]
TSP_A = [[1 if k in (i, j) else 0 for (i, j) in E] for k in range(n)]
TSP_b = [2] * n

TSP_res = linprog(TSP_c, TSP_A, TSP_b, bounds = bounds, method = 'simplex')
TSP_res

...in an attempt to reproduce this example (I don't know how to use panda by the way).

Updates:

• No bounds: some variables eaqual to 2, null function
• Bounds: 'bounds = [(0, 1)] * n' returns ValueError: Invalid input for linprog: unable to interpret bounds with this dimension tuple: (10, 2).
• Correct bounds: 'bounds = [(0, 1)] * (n*n)', variables equal to 1 or 0 but still null function + it seems only the diagonal (eg variables (i,j) = (0,0), (1,1)...) is equal to 1

Answer 1

This works:

d = [[distance_ij(coord_rad,i,j) for j in range(len(coord_rad))] for i in range(len(coord_rad))]

n = len(d)
E = [[i, j] for i in range(n) for j in range(n)]

TSP_A_eq = []

for k in range(n) :
    TSP_A_eq.append([1 if i == k and k != j else 0 for [i, j] in E]) # somme des arêtes sortantes
    TSP_A_eq.append([1 if k == j and i != k else 0 for [i, j] in E]) # somme des arêtes rentrantes

for i in range(n) :
    TSP_A_eq.append([1 if i == j else 0 for [i, j] in E]) # pas d'arête (i, i)

TSP_c = [d[i][j] for [i, j] in E]
TSP_b_eq = [1] * (2 * n) + [0] * n
bounds = [(0, 1)] * (n * n)

TSP_res = linprog(c=TSP_c, A_eq=TSP_A_eq, b_eq=TSP_b_eq, bounds=bounds, method='simplex', integrality=1)

Then there are the subtour elimination constraints...