Example #2

Minimize A Constrained Function

This is problem 6 in Chapter 10 of the book “Numerical Methods in Engineering with Python” by Jaan Kiusalaas.

The task is to minimize the function F(x,y) where x+y <= 1 and x >= 0.6

f(x,y) = (x-1)2 + (y-1)2
    where:    x+y <= 1
    and:      x >= 0.6

The parasol file begins very much as in Example 1, except that there is a new, second result variable called “sumXY” with a limit placed on it.

The setResultVariableLimits statement sets a high limit of 1.0 on “sumXY” in order to satisfy the x+y <= 1 constraint.

The parasol file begins as shown below, by importing parasol and defining the problem.

download func_opt_b.py

from parasol import *

PS = ParametricSoln(taskName="Minimize F(x,y) with x>=0.6 and x+y<=1",
     subtaskName="F(x,y)=(x-1)^2 + (y-1)^2",
     author="Charlie Taylor",
     constraintTolerance=0.001)


# design vars have:
#     name, value, minVal, maxVal, NSteps,  units,  description
PS.addDesVars(
    ['x',0.6,.6,1.,50,'','X value'],
    ['y',0.2,-0.2,0.4,50,'','Y value'],
    )

# result variables have:
#    name,      units,  description
PS.addResultVars(
    ['F','','Function Value'],
    ['sumXY','','Sum of X and Y'],
    )

PS.setResultVariableLimits(name="sumXY", hiLimit=1.)

The control routine is much like Example 1 except there is a second result variable to assign, “sumXY”.

# the following control routine ties together the system result variables
#  with the system design variables
def myControlRoutine(PS):
    # get current values of design variables
    x,y = PS("x","y")
    # set output variable values
    PS["F"] = (x-1.)**2 + (y-1.)**2
    PS["sumXY"] = x + y

The final task required for the setup are to give the ParametricSoln object the name of the control routine (myControlRoutine).

# need to tell system the name of the control routine
PS.setControlRoutine(myControlRoutine)

Again like Example 1, the statements below minimize “F”, save a summary, and make some plots to help visualize the results.

This example also introduces a carpet plot to display results.

# now optimize the system.
minimize(PS, figureOfMerit="F", desVars=[ 'x','y'], MaxLoop=500)

make2DParametricPlot(PS, sysParam="F", desVar="x",
    paramVar=["y",0.0,.1,.2,.3,.4,]  ,makeHTML=1, dpi=70, linewidth=2,
    ptData=None, ptLegend='', logX=0, logY=0)

makeCarpetPlot(PS, sysParam="F", xResultVar="sumXY",
    desVarL=[["x",0.6,0.8,1.0],["y",0.0,0.2,0.4]], angDesVarL=[40,0])

makeContourPlot(PS, sysParam="F", desVars=["x","y"])

# now save summary of system
PS.saveFullSummary()

The only remaining task is to tell parasol to close all files.

PS.close()  # finish up with output files

The output from the minimize operation is shown below.

Minimize F(x,y) with x>=0.6 and x+y<=1
F(x,y)=(x-1)^2 + (y-1)^2	ParametricSoln v0.1.7
by: Charlie Taylor	August 01, 2016

PRIOR TO MINIMIZE OPTIMIZATION

ParametricSoln: F(x,y)=(x-1)^2 + (y-1)^2 ====================== OPTIMIZATION DESIGN VARIABLES ======================= name value minimum maximum x 0.6 0.6 1 X value y 0.2 -0.2 0.4 Y value Figure of Merit: Function Value (F) = 0.8 <== Minimize ========================== OPTIMIZATION CONSTRAINTS ========================= name value minimum maximum sumXY 0.8 ------------ 1 Sum of X and Y ============================================================================

AFTER MINIMIZE OPTIMIZATION

ParametricSoln: F(x,y)=(x-1)^2 + (y-1)^2 ====================== OPTIMIZATION DESIGN VARIABLES ======================= name value minimum maximum x 0.6 0.6 1 X value y 0.4 -0.2 0.4 Y value Figure of Merit: Function Value (F) = 0.52 <== Minimize ========================== OPTIMIZATION CONSTRAINTS ========================= name value minimum maximum sumXY 1 ------------ 1 Sum of X and Y ============================================================================

Design Variable Summary
Design Variables (nominal values) Name Value Units Description y 0.4 Y value x 0.6 X value Result Variables Name Value Units Description Low Limit High Limit sumXY 1 Sum of X and Y --- <1 F 0.52 Function Value --- ---

Design Variable Summary
Design Variables (nominal values) Name Value Units Description y 0.4 Y value x 0.6 X value Result Variables Name Value Units Description Low Limit High Limit sumXY 1 Sum of X and Y --- <1 F 0.52 Function Value --- ---

Parametric Solutions parasol v0.1.7 contact: C. Taylor, cet@appliedpython.com