Multi-Objective Covariance Matrix Adaptation Evolution Strategy¶

The multi-objective covariance matrix adaptation evolution strategy (MO-CMA-ES) is one of the most powerful evolutionary algorithms for multi-objective real-valued optimization. In Shark, we provide a reference implementation of the algorithm (see MOCMA.h) that is integrated with the optimizer class hierarchy.

This tutorial illustrates applying the MO-CMA-ES to the DTLZ2() benchmark function. Please note that the methods presented here apply to all multi-objective optimizers available in the Shark library. That is, applying an optimizer to an objective function requires the following steps:

Instantiate and configure the objective function.
Instantiate the optimizer.
Configure the optimizer instance for the objective function instance.
Execute the optimizer until a termination criterion is fulfilled.

First of all, the following header files are required:

// Implementation of the MO-CMA-ES
#include <shark/Algorithms/DirectSearch/MOCMA.h>
// Access to benchmark functions
#include <shark/ObjectiveFunctions/Benchmarks/Benchmarks.h>

Next, an instance of the objective function is created and configured for a two-dimensional objective space and a three-dimensional search space, respectively:

shark::DTLZ2 dtlz2;
dtlz2.setNoVariables( 3 );
dtlz2.setNoObjectives( 2 );

Thereafter, the optimizer is instantiated and initialized for the objective function instance:

shark::detail::MOCMA mocma;
mocma.init( dtlz2 );

Subsequently, we declare a variable for storing the current approximation of the Pareto-optimal front:

std::vector<
    shark::TypedSolution<
        shark::DTLZ2::SearchPointType,
        shark::DTLZ2::ResultType
    >
> currentSolution;

In general, the result of an iteration of a multi-objective optimizer is a set of tuples consisting of the best known search point and its associated vector-valued fitness valuess.

Finally, we iterate the optimizer until the objective function instance has been evaluated 25000 times:

while( dtlz2.evaluationCounter() < 25000 ) {
    currentSolution = mocma.step( dtlz2 );
}

Running the example and visualizing the resulting Pareto-front approximation with the help of gnuplot will give you the following graphics:

Please see the file MOCMASimple.cpp for the complete source code of this tutorial.