DTLZ1

Here, we will show how to use HyperMapper to optimize the adapted DTLZ1 function presented in ParEGO. We look for minimizing the two values of this function given six parameters. The Pareto front of this function consists of all solutions where all but the first parameter are equal to 0.5.

The Objective Function

The values of the DTLZ1 function are defined as:

def dtlz1_function(x1, x2, x3, x4, x5, x6):
    x1 = X['x1']
    x2 = X['x2']
    x3 = X['x3']
    x4 = X['x4']
    x5 = X['x5']
    x6 = X['x6']
    g_sum = (x2 - 0.5)**2 + (x3 - 0.5)**2 + (x4 - 0.5)**2 + (x5 - 0.5)**2 + (x6 - 0.5)**2
    g_cos = math.cos(2*math.pi*(x2 - 0.5)) + math.cos(2*math.pi*(x3 - 0.5)) + math.cos(2*math.pi*(x4 - 0.5))\
          + math.cos(2*math.pi*(x5 - 0.5)) + math.cos(2*math.pi*(x6 - 0.5))
    g = 100*(5 + g_sum - g_cos)
    f1_value = 0.5*x1*(1 + g)
    f2_value = 0.5*(1 - x1)*(1 + g)

    output = {}
    output['f1_value'] = f1_value
    output['f2_value'] = f2_value

    return output

An example of this code can be found in dtlz1.py.

The JSON Configuration File

The json configuration file for this example is:

{
    "application_name": "dtlz1",
    "optimization_objectives": ["f1_value", "f2_value"],
    "optimization_iterations": 185,
    "design_of_experiment": {
        "doe_type": "standard latin hypercube",
        "number_of_samples": 65
    },
    "input_parameters" : {
        "x1": {
            "parameter_type" : "real",
            "values" : [0, 1]
        },
        "x2": {
            "parameter_type" : "real",
            "values" : [0, 1]
        },
        "x3": {
            "parameter_type" : "real",
            "values" : [0, 1]
        },
        "x4": {
            "parameter_type" : "real",
            "values" : [0, 1]
        },
        "x5": {
            "parameter_type" : "real",
            "values" : [0, 1]
        },
        "x6": {
            "parameter_type" : "real",
            "values" : [0, 1]
        }
    }
}

You can find this json in dtlz1_scenario.json.

Run HyperMapper

Remember to add the scripts folder to your $PYTHONPATH. In order to run this example, we use:

cd $HYPERMAPPER_HOME
python3 example_scenarios/synthetic/dtlz1/dtlz1.py

An example of stdout output can be found here.

The result of this script is a csv file called dtlz1_output_samples.csv.

Compute Pareto

After running HyperMapper, we can compute the Pareto front using the compute_pareto.py script:

python3 scripts/compute_pareto.py https://github.com/luinardi/hypermapper/blob/master/example_scenarios/synthetic/dtlz1/dtlz1_scenario.json

The result of this script is a csv file called dtlz1_output_pareto.csv.

Visualize Pareto

We can also visualize the Pareto front using the plot_pareto.py script:

python3 scripts/plot_pareto.py https://github.com/luinardi/hypermapper/blob/master/example_scenarios/synthetic/dtlz1/dtlz1_scenario.json

The result of this script is a pdf image dtlz1_output_pareto.pdf.