Discrete Optimization: Searching the Global Maximum

In this tutorial we show how to optimize a function with discrete input parameters along with continuous ones.

Problem Description

We want to minimize the following function:

The Objective Function

We create a folder model, and inside a file model.py, where we define the function that we want to optimize:

def model(d):
  npar = 10
  res = 0.0
  v = d["Parameters"]

  for i in range(npar):
    if( i == 0 or i == 1 or i == 3 or i == 6):
        res += pow( 10, 6.0*i/npar) * round(v[i]) * round(v[i])
    else:
        res += pow( 10, 6.0*i/npar) * v[i] * v[i]

  d["Evaluation"] = -res;

Then, in another file, for example run-cmaes.py, we start by importing the function we just defined (assuming model.py is in subfolder model relative to the current file), and creating an Experiment which we will configure,

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

import korali
e = korali.Experiment()

The Problem Type

We choose direct evaluation as problem type, set the objective function to our previously defined model(), and choose maximization as objective,

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

The Variables

We define 10 variables, of which four (\(x_1, x_2, x_4, x_7\)) are discrete. Also, we limit their domain to [-19, 21] each.

for i in range(10) :
  e["Variables"][i]["Name"] = "X" + str(i)
  e["Variables"][i]["Initial Mean"] = 1.0
  e["Variables"][i]["Lower Bound"]  = -19.0
  e["Variables"][i]["Upper Bound"]  = +21.0

# We set some of them as discrete.
e["Variables"][0]["Granularity"] = 1.0
e["Variables"][1]["Granularity"] = 1.0
e["Variables"][3]["Granularity"] = 1.0
e["Variables"][6]["Granularity"] = 1.0

The Solver

We choose the solver CMA-ES and set two termination criteria,

e["Solver"]["Type"] = "Optimizer/CMAES"
e["Solver"]["Population Size"] = 8
e["Solver"]["Termination Criteria"]["Min Value Difference Threshold"] = 1e-9
e["Solver"]["Termination Criteria"]["Max Generations"] = 5000

Output configuration

To redcue output frequency of result files and on the console we set

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

The Korali Engine Object

We create a Korali engine, and tell it to run the experiment we defined,

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

Running

We are now ready to run our example: ./run-cmaes.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.plot which visualizes the results found in folder _korali_result.