OpenMDAO Standard Library

The OpenMDAO standard library contains an assortment of useful plugins to the framework sorted into four catgories: components, drivers, factories, and traits.

Components

Drivers

The CONMIN Driver

CONMIN is a Fortran program written in subroutine form for the solution of linear or nonlinear constrained optimization problems. The basic optimization algorithm is the Method of Feasible Directions. If analytic gradients of the objective or constraint functions are not available, this information is calculated by finite difference. While the program is intended primarily for efficient solution of constrained problems, unconstrained function minimization problems may also be solved, and the conjugate direction method of Fletcher and Reeves is used for this purpose.

More information on CONMIN can be found in the CONMIN User’s Manual.

CONMIN has been included in the OpenMDAO standard library to provide users with a basic gradient-based optimization algorithm.

Basic Interface

The CONMIN code contains a number of different parameters and switches that are useful for controlling the optimization process. These can be subdivided into those parameters that will be used in a typical optimization problem and those that are more likely to be used by an expert user.

For the simplest possible unconstrained optimization problem, CONMIN needs just an objective function and one or more decision variables (design variables)

The OpenMDAO CONMIN driver can be from openmdao.main.api.

from openmdao.lib.api import CONMINdriver

Note that it can also be loaded by importing the CONMINdriver component from the standard library drivers namespace.

from openmdao.lib.drivers.conmindriver import CONMINdriver

Typically, CONMIN will be used as a driver in the top level assembly, though it can be also used in a subassembly as part of a nested driver scheme. Using the OpenMDAO script interface, a simple optimization problem can be set up as follows:

from openmdao.main.api import Assembly
from openmdao.lib.api import CONMINdriver

class EngineOptimization(Assembly):
    """ Top level assembly for optimizing a vehicle. """

    def __init__(self, directory=''):
        """ Creates a new Assembly containing a DrivingSim and an optimizer"""

        super(EngineOptimization, self).__init__(directory)

        # Create DrivingSim component instances
        self.add_container('driving_sim', DrivingSim())

        # Create CONMIN Optimizer instance
        self.add_container('driver', CONMINdriver())

This first section of code defines an assembly called EngineOptimization. This assembly contains a DrivingSim component and a CONMIN driver, both of which are created and added inside the __init__ function with add_container(). The objective function, design variables, onstraints, and any CONMIN parameters are also assigned in the __init__ function. The specific syntax for all of these is given below.

Both the objective function and the design variables are assigned via a StringRef variable. A StringRef is a string that points to some other OpenMDAO variable in the variable tree. There is only one objective function, but there can be multiple design variables which are assigned as a Python list.

# CONMIN Objective
self.driver.objective = 'driving_sim.accel_time'

# CONMIN Design Variables
self.driver.design_vars = ['driving_sim.spark_angle',
                                       'driving_sim.bore' ]

Note that all input parameters for the CONMIN driver are assigned via self.driver.

These StringRef variables must point to something that can be seen in the scope of the CONMIN driver. In other words, if an assembly contains a CONMIN driver, the objective function and design variables cannot be located outside of that assembly. Also, each design variable must point to a component input. During the optimization process, the design variables are modified, and the relevant portion of the model is executed to evaluate the new objective. Note that it is generally not possible to connect more than one driver to an available input.

Additionally, the objective function must always be either an output from a component or a function of available component outputs:

# CONMIN Objective = Maximize weighted sum of EPA city and highway fuel economy
self.driver.objective = '-(.93*driving_sim.EPA_city + 1.07*driving_sim.EPA_highway)'

In this example, the objective is to maximize the weighted sum of two variables. The equation must be constructed using valid Python operators. All variables in the function are expressed in the scope of the local assembly that contains the CONMIN driver.

More realistically, optimization problems usually have constraints. There are two types of constrains in CONMIN – ordinary constraints, which are expressed as functions of the design variables, and side constraints, which are used to bound the design space (i.e., specify a range for each design variable).

Side constraints are defined using the lower_bounds and upper_bounds parameters:

self.driver.lower_bounds = [-50, 65]
self.driver.upper_bounds = [10, 100]

The size of these lists must be equal to the number of design variables or OpenMDAO will raise an exception. Similarly, the upper bound must be greater than the lower bound for each design variable.

Constraints are equations (or inequalities) much like the objective function, so they are also constructed from the available OpenMDAO variables using Python mathematical syntax. The constraints parameter is a list of inequalities that are defined to be satisfied when they return a negative value or zero, and violated when they return a positive value.

self.driver.constraints = ['driving_sim.stroke - driving_sim.bore']

Note that any equation can also be expressed as an inequality.

Controlling the Optimization

It is often necessary to control the convergence criteria for an optimization. The CONMIN driver allows the user to control both the number of iterations before termination as well as the convergence tolerance (both absolute and relative).

The maximum number of iterations is specified by setting the itmax parameter. The default value is 10.

self.driver.itmax = 30

The convergence tolerance is controlled with delfun and dabfun. Delfun is the absolute change in the objective function to indicate convergence (i.e., if the objective function changes by less than delfun, then the problem is converged). Similarly, dabfun is the relative change of the objective function with respect to the value at the previous step. Note that dabfun has a hard-wired minimum of 1e-10 in the Fortran code, and delfun has a minimum of 0.0001.

self.driver.dabfun = .001
self.driver.delfun = .1

All of these convergence checks are always active during optimization. The tests are performed in the following sequence:

  1. Check number of iterations
  2. Check absolute change in objective
  3. Check relative change in objective
  4. Reduce constraint thickness for slow convergence

There is also a parameter to control how many iterations the convergence tolerance should be checked before terminating the loop. This is done with the itrm parameter, whose default value is 3.

self.driver.itrm = 3

CONMIN can calculate the gradient of both the objective functions and of the constraints using a finite difference approximation. This is the current default behavior of the OpenMDAO driver. The CONMIN code can also accept user-calculated gradients, but these are not yet supported in OpenMDAO. There are two parameters that control the step size used for numerically estimating the local gradient.

self.driver.fdch = .0001
self.driver.fdchm = .0001

The fdchm parameter is the minimum absolute step size that the finite difference will use, and fdch is the step size relative to the design variable.

Note

The default values of fdch and fdchm are set to 0.01. This may be too low for some problems and will manifest itself by converging to a value that is not the minimum. It is important to evaluate the scale of the objective function around the optimum so that these can be chosen well.

For certain problems, it is desirable to scale the inputs. There are several scaling options available, as summarized here:

Value Result
nscal < 0 User-defined scaling with the vector in scal
nscal = 0 No scaling of the design variables
nscal > 0 Scale the design variables every NSCAL iterations. Please see the CONMIN user’s manual for additional notes about using this option

The default setting is nscal=0 for no scaling of the design variables. The nscal parameter can be set to a negative number to turn on user-defined scaling. When this is enabled, the array of values in the vector scal is used to scale the design variables.

self.driver.scal = [10.0, 10.0, 10.0, 10.0]
self.driver.nscal = -1

Note that there need to be as many scale values as there are design variables.

Finally, the iprint parameter can be used to turn on the display of diagnostic messages inside of CONMIN. These messages are currently sent to the standard output.

self.driver.iprint = 0

Higher positive values of iprint turn on the display of more levels of output, as summarized below.

Value Result
iprint = 0 All output is suppressed
iprint = 1 Print initial and final function information
iprint = 2 Debug level 1: All of the above plus control parameters
iprint = 3 Debug level 2: All of the above plus all constraint values, number of active/violated constraints, direction vectors, move parameters, and miscellaneous information
iprint = 4 Complete debug: All of the above plus objective function gradients, active and violated constraint gradients, and miscellaneous information
iprint = 5 All of above plus each proposed design vector, objective and constraints during the one-dimensional search
iprint = 101 All of above plus a dump of the arguments passed to subroutine CONMIN

Advanced Options

The following options exercise some of the more advanced capabilities of CONMIN. The details given here briefly summarize the effects of these parameters; more information is available in the CONMIN User’s Manual.

icndir – Conjugate direction restart parameter. For an unconstrained problem (no side constraints either), Fletcher-Reeves conjugate direction method will be restarted with the steepest descent direction every ICNDIR iterations. If ICNDIR = 1, only the steepest descent will be used. Default value is the number of design variables + 1.

Constraint Thickness – CONMIN gives four parameters for controlling the thickness of constraints – ct, ctmin, ctl, and ctlmin. Using these parameters essentially puts a tolerance around a constraint surface. Note that ct is used for general constraints, and ctl is just used for linear constraints. A wide initial value of the constraint thickness is desirable for highly nonlinear problems so that when a constraint becomes active, it tends to remain active, thus reducing the zigzagging problem. The values of ct and ctl adapt as the problem converges, so the minima can be set with ctl and ctlmin.

theta – Mean value of the push-off factor in the method of feasible directions. A larger value of theta is desirable if the constraints are known to be highly nonlinear, and a smaller value may be used if all constraints are known to be nearly linear. The actual value of the push-off factor used in the program is a quadratic function of each constraint (G(J)), varying from 0.0 for G(J) = ct to 4.0*theta for G(J) = ABS(ct). A value of theta = 0.0 is used in the program for constraints which are identified by the user to be strictly linear. Theta is called a push-off factor because it pushes the design away from the active constraints into the feasible region. The default value is usually adequate. This is only used for constrained problems.

phi – Participation coefficient, used if a design is infeasible (i.e., one or more violated constraints). Phi is a measure of how hard the design will be “pushed” towards the feasible region and is, in effect, a penalty parameter. If in a given problem, a feasible solution cannot be obtained with the default value, phi should be increased, and the problem run again. If a feasible solution cannot be obtained with phi = 100, it is probable that no feasible solution exists. The default value of 5.0 is usually adequate. This is only used for constrained problems.

linobj – Set this to 1 if the objective function is known to be linear.

PyEvolve

The Case Iterator

Factories

Traits