Individual Design Feasible (IDF)

Next, we will look at how to set up the Individual Design Feasible (IDF) architecture using the Sellar problem. In IDF, the coupling between the disciplines is removed, and the input coupling variables are added to the optimizer’s design variables. The algorithm calls for two new equality constraints that constrain to zero the residual error between the coupling variable output by the optimizer and the coupling variable output by the components. This assures that the solution is a feasible coupling, though it is achieved through the optimizer’s additional effort instead of a solver. The data flow for IDF is illustrated in the following diagram:

Data Flow for IDF

IDF needs only one driver, so there is just one workflow. The broadcaster and the two disciplines are executed sequentially.

Iteration Hierarchy for IDF

Next, we will create the SellarIDF assembly. First, all of our components are instantiated and the workflow is defined.

from openmdao.examples.mdao.disciplines import SellarDiscipline1, \
                                               SellarDiscipline2
from openmdao.examples.mdao.broadcaster import Broadcaster

from openmdao.main.api import Assembly, set_as_top
from openmdao.lib.drivers.api import CONMINdriver


class SellarIDF(Assembly):
    """ Optimization of the Sellar problem using IDF"""

    def __init__(self):
        """ Creates a new Assembly with this problem

        Optimal Design at (1.9776, 0, 0)

        Optimal Objective = 3.18339"""

        # pylint: disable-msg=E1101

        super(SellarIDF, self).__init__()

        # create Optimizer instance
        self.add('driver', CONMINdriver())

        # Disciplines
        self.add('bcastr', Broadcaster())
        self.add('dis1', SellarDiscipline1())
        self.add('dis2', SellarDiscipline2())

        # Driver process definition
        self.driver.workflow.add(['bcastr', 'dis1', 'dis2'])

        # Make all connections
        self.connect('bcastr.z1','dis1.z1')
        self.connect('bcastr.z1','dis2.z1')
        self.connect('bcastr.z2','dis1.z2')
        self.connect('bcastr.z2','dis2.z2')

We’ve also hooked up our data connections. Only the design variables that are shared by both components need to be connected to the broadcaster.

All that is left to do is set up the CONMIN optimizer.

# Optimization parameters
self.driver.add_objective('(dis1.x1)**2 + bcastr.z2 + dis1.y1 + math.exp(-dis2.y2)')

self.driver.add_parameter('bcastr.z1_in', low = -10.0, high=10.0)
self.driver.add_parameter('bcastr.z2_in', low = 0.0,   high=10.0)
self.driver.add_parameter('dis1.x1',      low = 0.0,   high=10.0)
self.driver.add_parameter('dis2.y1',      low = 3.16,  high=10.0)
self.driver.add_parameter('dis1.y2',      low = -10.0, high=24.0)

self.driver.add_constraint('(dis2.y1-dis1.y1)**3 < 0')
self.driver.add_constraint('(dis1.y1-dis2.y1)**3 < 0')
self.driver.add_constraint('(dis2.y2-dis1.y2)**3 < 0')
self.driver.add_constraint('(dis1.y2-dis2.y2)**3 < 0')
self.driver.iprint = 0
self.driver.itmax = 100
self.driver.fdch = .003
self.driver.fdchm = .003
self.driver.delfun = .0001
self.driver.dabfun = .00001
self.driver.ct = -.01
self.driver.ctlmin = 0.001

Notice that the coupling variables are included as optimizer parameters. We also introduce the CONMIN parameter ct, which is the constraint thickness for nonlinear constraints. Our constraints are nonlinear, but note that any constraint that involves a component output is most likely a nonlinear constraint because outputs are usually nonlinear functions of the design variables.

Since CONMIN doesn’t support equality constraints, we have to fall back on a trick where we replace it with an equivalent pair of inequality constraints. For example, if we want to constrain x=2, we could constraint x<=2 and x>=2 and let the optimizer converge to a solution where both constraints are active. Stability may be questionable for such a method, so it is always advisable to use an optimizer that has equality constraints rather than trying to squeeze a solution out of an optimizer this way. In particular, be careful about trying a fancier solution such as constraining abs(dis2.y1-dis1.y1)<=0. This nonlinear constraint has a discontinuous slope, and CONMIN won’t handle that constraint very well. Here, we take (dis2.y1-dis1.y1) and turn it into a cubic expression, which seemed to make the problem a little less sensitive to changes in the computational environment (32 bit vs 64 bit, etc.)

This problem is contained in sellar_IDF.py. Executing it at the command line should produce output that resembles this:

$ python sellar_IDF.py
CONMIN Iterations:  10
Minimum found at (1.976427, 0.000287, 0.000000)
Couping vars: 3.156521, 3.754359
Minimum objective:  3.18022323743
Elapsed time:  0.200541973114 seconds