Constrained Optimization: Searching the Global Maximum¶

In this tutorial we show how to solve a constrained optimization problem CEC 2006 Test Problem g09), defined as: Find \(x^\star = \mathop{\arg\min}\limits_{x} \,\,\, f(x) \,,\) under the constraints \(g_i(x) \leq 0.\)

Problem Description¶

We want to solve the problem:

under the four constraints \(g_i(x)\):

The Objective Function¶

Create a folder named model. Inside, create a file with name model.py and paste the following code,

#!/usr/bin/env python

def g09( k ):

  d = k["Parameters"]
  res = (d[0] - 10.0)**2 + 5.0 * (d[1] - 12.0)**2           \
        + d[2]**4  + 3.0 * (d[3] - 11.0)**2                 \
        + 10.0 * d[4]**6 + 7.0 * d[5]**2 + d[6]**4.      \
        - 4.0 * d[5] * d[6] - 10.0 * d[5] - 8.0 * d[6];

  k["Evaluation"] = -res;

This computational model represents our objective function.

For the constraints, add the following code in the same file,

def g1(k):
  v = k["Parameters"]
  k["Evaluation"] = -127.0 + 2 * v[0] * v[0] + 3.0 * pow(v[1], 4) + v[2] + 4.0 * v[3] * v[3] + 5.0 * v[4]


def g2(k):
  v = k["Parameters"]
  k["Evaluation"] = -282.0 + 7.0 * v[0] + 3.0 * v[1] + 10.0 * v[2] * v[2] + v[3] - v[4]


def g3(k):
  v = k["Parameters"]
  k["Evaluation"] = -196.0 + 23.0 * v[0] + v[1] * v[1] + 6.0 * v[5] * v[5] - 8.0 * v[6]

def g4(k):
  v = k["Parameters"]
  k["Evaluation"] = 4.0 * v[0] * v[0] + v[1] * v[1] - 3.0 * v[0] * v[1] + 2.0 * v[2] * v[2] + 5.0 * v[5] - 11.0 * v[6]

Optimization with (C)CMA-ES¶

First, open a file and import the korali module

#!/usr/bin/env python3
import korali

Import the computational model,

import sys
sys.path.append('./model')
from model import *
from constraints import *

The Korali Object¶

Next we construct a korali.Experiment object,

e = korali.Experiment()

Add the objective function and the constraints in the Korali object,

e["Problem"]["Objective Function"] = g09
e["Problem"]["Constraints"] = [ g1, g2, g3, g4 ]

The Problem Type¶

Then, we set the type of the problem to Direct Evaluation

e["Problem"]["Type"] = "Evaluation/Direct/Basic"
e["Problem"]["Objective"] = "Maximize"

The Variables¶

We add 7 variables to the experiment and set their domain,

for i in range(7) :
  e["Variables"][i]["Name"] = "X" + str(i)
  e["Variables"][i]["Lower Bound"] = -10.0
  e["Variables"][i]["Upper Bound"] = +10.0

The Solver¶

We choose the solver CMA-ES,

e["Solver"]["Type"] = "Optimizer/CMAES"

Then we set a few parameters for CCMA-ES,

e["Solver"]["Is Sigma Bounded"] = True
e["Solver"]["Population Size"] = 32
e["Solver"]["Viability Population Size"] = 4
e["Solver"]["Termination Criteria"]["Max Value"] = -680.630057374402 - 1e-4
e["Solver"]["Termination Criteria"]["Max Generations"] = 500

For a detailed description of CCMA-ES settings see CMAES.

We configure output settings,

e["File Output"]["Frequency"] = 50
e["Console Output"]["Frequency"] = 50

Finally, we need to create a Korali Engine object add a call to its run() routine, to start the engine.

k = korali.Engine()
k.run(e)

Running¶

We are now ready to run our example:./run-ccmaes.py

The results are saved in the folder _korali_result/.

Plotting¶

You can see the results of CMA-ES by running the command, python3 -m korali.plotter