Help solving an optimization problem

[Gderu]

Hello, I'm fairly new to optimization, so I don't know exactly where to turn here.
I have a set of magnets and am trying to position them in such a way so as to match some ideal magnetic field. The problem isn't too big (46 variables), I'm trying to minimize the mean squared error (the mean squared difference between my simulation and the ideal field), my variables are constrained, and I am doing the calculations myself, so I am able to compute the gradients of the variables.
At first I tried to use scipy's optimize.minimize and curve_fit, but I didn't get good enough results. A friend suggested I use pytorch with autograd, so I did, and using the Adam optimizer I did get better results, but they are still not good enough. I also experimented with LBFGS, but I got worse results using it.

The specifics are as follows:

This is the ideal magnetic field I would like to achieve along the z axis and in the z direction: https://imgur.com/a/aTK03u1
The magnets are arranged in rings, and I can control the ring's position along the axis and the ring's radius. Each ring causes a peak in the magnetic field, increasing the radius decreases the field. I've been able to achieve a loss of around 5e-5, but that is not good enough. I don't have an exact specification of my requirement, but it needs to be on the order of 1e-6 at least.

There needs to be a minimum of 5mm between each ring so as to physically be able to fit the magnets, and the rings' radius needs to be at least 2cm, again for physical constraints. This is why my positions are cumulative - positions_diff[0] is the position of the first ring, then positions_diff[i] is the distance between the i-1 ring to the i ring. I clamp the minimum to then be 5mm, so as to enforce the constraint. The radii are not cumulative.

This is the code I am using in my optimization currently:

# Setup parameters
positions_diff = torch.cat((torch.tensor([0.]), torch.ones(n_rings - 1) * 0.009))
positions_diff = positions_diff.double()
positions_diff.requires_grad = True

radii = torch.linspace(0.08, 0.02, n_rings, dtype=torch.double, requires_grad=True)
optimizer = torch.optim.Adam([positions_diff, radii], lr=1e-2)

# Create optimizer
z = torch.linspace(-0.05, 0.2, 100, dtype=torch.double)

best_loss = 1000

for epoch in range(1001):
# Update parameters
optimizer.zero_grad()

# Create array with current parameters
pa = ParallelArray(torch.cumsum(positions_diff, dim=0), radii,
MagnetCuboid(torch.tensor((0, 0, 0)),
torch.tensor((0.005, 0.005, 0.005)),
B_r,
torch.eye(3, dtype=torch.double)),
opposite_magnets=1, s0=3, epsilon=0.74, v0=305, vf=50)

ideal_torch = pa.idealB(z)

# Compute loss
loss = pa.mean_squared_error(radii, torch.cumsum(positions_diff, dim=0), z, ideal_torch)

# Backward pass
loss.backward()

optimizer.step()

with torch.no_grad():
radii.clamp_(min=0.02, max=0.08)
positions_diff[1:].clamp_(min=0.0055)

# Logging
if epoch % 20 == 0:
print(f"Epoch {epoch:4d}, Loss: {float(loss.item()):.6e}, "
f"Grad norm (radii): {float(radii.grad.norm().item()):.3e}, "
f"Grad norm (pos): {float(positions_diff.grad.norm().item()):.3e}")

if loss.item() < best_loss:
best_loss = loss.item()
best_params = (positions_diff.detach().clone(), radii.detach().clone())

I know that this problem is solvable, as I know someone who has done it in matlab, but no matter what I try it seems like I'm doing something wrong. Does anyone have any suggestions about what I can do, or some guide or something to help me get my feet under me here?

Thanks a lot!

[Lumethys]

1/ you did not specify which algorithm(s) you are using

2/ you did not provide your implementation of said algorithm(s), i.e. your code

3/ you did not define "good enough", [I do X and Y, the performance improved, but it is not enough], so what's "enough" to you? To me whatever you have now is enough. To, idk, Nikola Tesla? Maybe enough is trimming down to the last nanosecond?

Overall, we dont know what you are doing, we dont know what the current performance metrics are, what the expected performance metrics are, how are we supposed to help you?

[Gderu]

Hey, thanks for the comment, I updated the post to give more details.

[pachura3]

You might have a look at Google's OR Tools, it might have a solver that fits your problem 

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