What's wrong with my Hopfield Neural Network solution to the Traveling Salesman Problem?

Question

First off, this is homework. I think it's clear I've made an effort and I'm looking for hints, not code.

The problem is the following. The equation of operation has four components for altering a given neuron.

• A) One part to ensure each city is visited at most once.
• B) One to ensure each position (first, second, third, etc) has at most one city.
• C) One part to ensure that the total number of active neurons is equal to the number of cities.
• D) One part to minimize distance.

If I weight D heavily enough that it has any effect, the network settles on an invalid tour (for example, visit A, D, nowhere, E, C). I can, however, deweight D and the code will find solutions, but not those with minimal distance.

I'd be extremely grateful for any advice, I've been banging my head against the keyboard for a while. The code should be understandable by anyone familiar which solving the TSP with a Hopfield network.

Das Code:

%parameters
n=5;
theta = .5;
u0 = 0.02;
h = .1;
limit = 2000;

%init u
u=zeros(n,n);
uinit = -u0/2*log(n-1); %p94 uINIT = - u0/2 * ln(n-1) 
for i=1:n
    for j=1:n
        u(i,j) = uinit * (1+rand()*0.2-0.1); %add noise [-0.1*uInit 0.1*uINIT]
    end
end 

%loop
for index=1:limit
    i = ceil(rand()*n);
    k = ceil(rand()*n);

    %runge kutta
    k1 = h*du(u,i,k,0);
    k2 = h*du(u,i,k, k1/2);
    k3 = h*du(u,i,k, k2/2);
    k4 = h*du(u,i,k, k3);
    u(i,k) = u(i,k) + (k1 + 2*k2 + 2*k3 + k4)/6;
end

Vfinal = hardlim(V(u)-theta)

du()

function  out=du(u,X,i,c)

dist = [0, 41, 45, 32, 32;
        41, 0, 36, 64, 54;
        45, 36, 0, 76, 32;
        32, 64, 76, 0, 60;
        32, 54, 32, 60, 0];

t = 1;
n = 5;
A = 10;
B = 10;
C = 10;
D = .0001;


AComp = A*sum(V(u(X,:))) - A*V(u(X,i));
BComp = B*sum(V(u(:,i))) - B*V(u(X,i));
CComp = C*(sum(sum(V(u)))-n);

DComp = 0;
before = i-1;
after = i+1;
if before == 0
    before = 5;
end
if after == 6
    after = 1;
end
for Y=1:5
    DComp = DComp + dist(X,Y) * (V(u(Y,after)) + V(u(Y,before)));
end
DComp = DComp * D;

out = -1*(u(X,i)+c)/t - AComp - BComp - CComp - DComp;

V()

function  out=V(u)
u0 = 0.02;
out = (1 + tanh(u/u0))/2;

Answer 1

I have never tried solving the TSP with a neural network, but I have found that it solves very well, and very quickly, taking a genetic approach.

I have done many neural network projects, though, and I would guess that since the TSP can, in general, have many solutions over a single network (of cities), that the neural network could be dragged back and forth between solutions, never really successfully converging on any one.

John R. Doner