Individual Design Feasible (IDF)

Next, we will look at how to set up the Individual Design Feasible (IDF) architecture for the Sellar problem. In IDF, the direct 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 enforce the coupling between the disciplines. This ensures 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 where the two disciplines are executed sequentially. From the perspective of the iteration hierarchy, IDF is extremely simple.

Iteration Hierarchy for IDF

To implement IDF, we create the SellarIDF assembly. First, all of our components are instantiated and the workflow is defined.

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

from openmdao.lib.optproblems import sellar

class SellarIDF(Assembly): #TEST
    """ 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"""

        super(SellarIDF, self).__init__()

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

        # Disciplines
        self.add('dis1', sellar.Discipline1())
        self.add('dis2', sellar.Discipline2())

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

That’s all it takes to setup the workflow for IDF. All that is left to do is set up the CONMIN optimizer. In the code below, pay attention to how we handle the global design variables z1 and z2. We set them up the same way we did for the MDF architecture. However, unlike the MDF, the coupling variables are also 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.

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

#Global Design Variables
self.driver.add_parameter(('dis1.z1','dis2.z1'), low = -10.0, high=10.0)
self.driver.add_parameter(('dis1.z2','dis2.z2'), low = 0.0,   high=10.0)

#Local Design Variables and Coupling Variables
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

Technically, IDF requires the use of equality constraints to enforce coupling between the disciplines. 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. Working with two inequalities is a bit trickier though, because it can introduce some instability to the optimizer and affect its final solution.

You might consider trying a fancier solution such as constraining abs(dis2.y1-dis1.y1)<=0. Be careful though, because this nonlinear constraint has a discontinuous slope, and CONMIN won’t handle that very well. After some experimentation, we found that cubing the difference between the coupling variables, i.e., (dis1.y1-dis2.y1)**3, seemed to make CONMIN happy and helped convergence.

When you put it all together, you get sellar_IDF.py. Once again, we added a small amount of code at the end to execute and then print the results of the IDF optimization.

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

from openmdao.lib.optproblems import sellar


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"""

        super(SellarIDF, self).__init__()

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

        # Disciplines
        self.add('dis1', sellar.Discipline1())
        self.add('dis2', sellar.Discipline2())

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


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

        #Global Design Variables
        self.driver.add_parameter(('dis1.z1','dis2.z1'), low = -10.0, high=10.0)
        self.driver.add_parameter(('dis1.z2','dis2.z2'), low = 0.0,   high=10.0)

        #Local Design Variables and Coupling Variables
        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


if __name__ == "__main__":
    import time

    prob = SellarIDF()
    set_as_top(prob)

    # pylint: disable-msg=E1101

    prob.dis1.z1 = prob.dis2.z1 = 5.0
    prob.dis1.z2 = prob.dis2.z2 = 2.0
    prob.dis1.x1 = 1.0
    prob.dis2.y1 = 3.16

    tt = time.time()
    prob.run()

    print "\n"
    print "CONMIN Iterations: ", prob.driver.iter_count
    print "Minimum found at (%f, %f, %f)" % (prob.dis1.z1, \
                                             prob.dis2.z2, \
                                             prob.dis1.x1)
    print "Couping vars: %f, %f" % (prob.dis1.y1, prob.dis2.y2)
    print "Minimum objective: ", prob.driver.eval_objective()
    print "Elapsed time: ", time.time()-tt, "seconds"

Executing this 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