OpenMDAO Standard Library

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

Components

Drivers

The CONMIN Driver

CONMIN is a Fortran program written as a subroutine to solve 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. 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. (In the simple tutorial in the User Guide, CONMIN is used for an unconstrained and a constrained optimization.)

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 just needs an objective function and one or more decision variables (design variables)

The OpenMDAO CONMIN driver can be imported from openmdao.lib.api.

from openmdao.lib.api 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):
        """ Creates a new Assembly containing a DrivingSim and an optimizer"""

        super(EngineOptimization, self).__init__()

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

        # Create CONMIN Optimizer instance
        self.add('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, constraints, 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 an Expression variable. An Expression 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' ]

These Expression variables must point to something that can be seen in the scope of the asssembly that contains 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. 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.

There are two types of constraints 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 that are 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']

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

The number of successive iterations that the convergence tolerance should be checked before terminating the loop can also be specified 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. Two parameters control the step size used for numerically estimating the local gradient: fdch and fdchm. 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.

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

Note

The default values of fdch and fdchm are set to 0.01. This may be too large 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. Several scaling options are 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

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

If your problem uses linear constraints, you can improve the efficiency of the optimization process by designating those that are linear functions of the design variables as follows:

self.driver.constraints = ['driving_sim.stroke - driving_sim.bore',
                           '1.0 - driving_sim.stroke * driving_sim.bore']
self.cons_is_linear = [1, 0]

If cons_is_linear is not specified, then all the constraints are assumed to be nonlinear. Note that the original CONMIN parameter for this is ISC.

Finally, the iprint parameter can be used to display 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 used only 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 used only 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. Phi is used only for constrained problems.

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

Genetic

Genetic is a driver which performs optimization using a genetic algorithm based on Pyevolve. Genetic is a global optimizer and is ideal for optimizing problems with integer or discrete design variables because it is a non-derivative based optimization method.

Genetic can be used in any simulation by importing it from openmdao.lib.api:

from openmdao.lib.api import Genetic

Design Variables

Public variables are added to Genetic and become design variables. Genetic will vary the set of design variables to search for an optimum. Genetic supports three public variable types: Float, Int, and Enum. These public variable types can be used as design variables in any optimization.

You add design variables to Genetic using the add_parameter method.

from openmdao.main.api import Assembly,Component, set_as_top
from openmdao.lib.api import Genetic
from openmdao.lib.api import Float,Int,Enum

class SomeComp(Component):
    """Arbitrary component with a few public variables, but which does not really do
    any calculations"""

    w = Float(0.0,low=-10,high=10,iotype="in")

    x = Float(0.0,low=0.0,high=100.0,iotype="in")
    y = Int(10,low=10,high=100,iotype="in")
    z = Enum([-10,-5,0,7],iotype="in")

class Simulation(Assembly):
    """Top Level Assembly used for simulation"""

    def __init__(self):
        """Adds the Genetic driver to the assembly"""

        super(Simulation,self).__init__()

        self.add('optimizer',Genetic())
        self.add('comp',SomeComp())

        self.optimizer.add_parameter('comp.x')
        self.optimizer.add_parameter('comp.y')
        self.optimizer.add_parameter('comp.z')

top = Simulation()
set_as_top(top)

In the above example, three design variables were added to the optimizer. The optimizer figures out for itself what type of variable it is and behaves appropriately. In all three cases, since no low or high arguments were provided, the optimizer will use the values from the metadata provided in the variable deceleration.

For comp.x the optimizer will try floats between 0.0 and 100.0. For comp.y the optimizer will try integers between 10 and 100. For comp.z the optimizer will pick from the list of allowed values: [-10,-5,0,7].

You can override the low and high values from the metadata if you want the optimizer to use a different range instead of the default.

top.optimizer.add_parameter('comp.w',low=5.0,high=7.0)

Now, for comp.x the optimizer will only try values between 5.0 and 7.0. Note that low and high are only applicable to Float and Int public variables. For Enum public variables, low and high are not applicable.

Configuration

When setting the objective attribute you can specify a single public variable or a more complex function, such as

top.optimizer.objective = "comp.x"

or

top.optimizer.objective = "2*comp.x+comp.y+3*comp.z"

In the second example above, a more complex objective was created where the overall objective was a weighted combination of comp.x, comp.y, and comp.z.

To set the optimizer to either minimize or maximize your objective, you set the opt_type attribute of the driver to “minimize” or “maximize.”

top.optimizer.opt_type = "minimize"

You can control the size of the population in each generation and the maximum number of generations in your optimization with the population_size and generations attributes.

top.optimizer.population_size = 80
top.optimizer.generations = 100

As you increase the population size, you are effectively adding diversity in to the gene pool of your optimization. A large population means that a larger number of individuals from a given generation will be chosen to provide genetic material for the next generation. So there is a better chance that weaker individuals will pass on their genes. This diversity helps to ensure that your optimization will find a true global optimum within the allowed design space. However, it also serves to slow down the optimization because of the increased number of function evaluations necessary for each generation.

Picking an appropriate value for the maximum number of generations will depend highly on the specifics of your problem. Setting this number too low will likely prevent the optimization from converging on a true optimum. Setting it too high will help you find the true optimum, but you may end up wasting the computation time on later generations where the optimum has been found.

You can further control the behavior of the genetic algorithm by setting the crossover_rate, mutation_rate, selection_method, and elitism attributes. These settings will allow you to fine-tune the convergence of your optimization to achieve the desired result; however, for many optimizations the default values will work well and won’t need to be changed.

The crossover_rate controls the rate at which the crossover operator gets applied to the genome of a set of individuals who are reproducing. The allowed values are between 0.0 and 1.0. A higher rate will mean that more of the genes are swapped between parents. The result will be a more uniform population and better searching of the design space. If the rate is set too high, then it is likely that stronger individuals could be lost to churn.

top.optimizer.crossover_rate = 0.9

The mutation_rate controls how likely any particular gene is to experience a mutation. A low, but non-zero, mutation rate will help prevent stagnation in the gene pool by randomly moving the values of genes. If this rate is set too high, the algorithm basically degrades into a random search through the design space. The allowed values are between 0.0 and 1.0.

top.optimizer.mutation_rate = .02

In a pure genetic algorithm, it is possible that your best performing individual will not survive from one generation to the next due to competition, mutation, and crossover. If you want to ensure that the best individual survives in tact from one generation to the next, then turn on the elitism flag for your optimization. This will ensure that the best individual is always copied to the next generation no matter what.

top.optimizer.elitism = True

A number of different commonly used selection algorithms are available. The default algorithm is the Roulette Wheel Algorithm, but Tournament Selection, Rank Selection, and Uniform Selection are also available. The selection_method attribute allows you to select the algorithm; allowed values are: “roulette_wheel,” “tournament,” “rank,” and “uniform.”

top.optimizer.selection_method="rank"

The Case Iterator

Todo

Case Iterator documentation

Factories

Traits