OpenMDAO 1.2.0 implicit component

Question

I new to OpenMDAO and I'm still learning how to formulate the problems. For a simple example, let's say I have 3 input variables with the given bounds:

1 <= x <= 10
0 <= y <= 10
1 <= z <= 10

and I have 4 outputs, defined as:

f1 = x * y
f2 = 2 * z
g1 = x + y - 1
g2 = z

my goal is to minimize f1 * g1, but enforce the constraint f1 = f2 and g1 = g2. For example, one solution is x=3, y=4, z=6 (no idea if this is optimal).

For this simple problem, you can probably just feed the output equality constraints to the driver. However, for my actual problem it's hard to find an initial starting point that satisfy all the constraints, and as the result the optimizer failed to do anything. I figure maybe I could define y and z as states in an implicit component and have a nonlinear solver figure out the right values of y and z given x, then feed x to the optimization driver.

Is this a possible approach? If so, how will the implicit component look like in this case? I looked at the Sellar problem tutorial but I wasn't able to translate it to this case.

Justin Gray Justin Gray · Accepted Answer · 2015-10-21T11:47:32

You could create an implicit component if you want. In that case, you would define an apply_linear method in your component. That is done with the sellar problem here.

In your case since you have a 2 equation set of residuals which are both dependent on the state variables, I suggest you create a single array state variable of length 2, call it foo (I used a new variable to avoid any confusion, but name it whatever you want!). Then you will define two residuals, one for each element of the residual array of the new state variable.

Something like:

resids['foo'][0] = params['x'] * unknowns['foo'][0] - 2 * unknowns['foo'][1]
resids['foo'][1] = params['x'] + unknowns['foo'][0] - 1 - unknowns['foo'][1]

If you wanted to keep the state variable names separate you could, and it will still work. You'll just have to arbitrarily assign one residual equation to one variable and one to the other.

Then the only thing left is to add a non linear solver to the group containing your implicit component and it should work. If you choose to use a newton solver, you'll either need to set fd_options['force_fd'] = True or define derivatives of your residuals wrt all params and state variables.

OpenMDAO 1.2.0 implicit component

1 Answers