# Creating Multi-Objective Benchmarks with Shark¶

This is short tutorial on how to use Shark to develop a simple Benchmark. It is intended to give a starting point to develop experimental designs. This tutorial is based on the MOCMAExperiment.cpp, which includes a complete implementation of this tutorial.

Shark comes with a basic set of multi-objective algorithms including the reference implementation of the MO-CMA-ES. Moreover, Shark includes a big set of standard multi-objective benchmark functions. The best known are the ZDT, DTLZ, IHR and LZ families of functions. All benchmarks are located in “ObjectiveFunctions/Benchmarks”.

In the following we will describe a standard setup to evaluate the MO-CMA-ES on some of the ZDT-functions. It is not an easy task to evaluate the quality of a solution of a multi-objective function - we have to evaluate how good the points approximate the optimal pareto-front. Instead of doing this directly, we will measure the hypervolume covered by the points with respect to a reference point. Optimal results are known for some benchmarks. We will repeatedly optimize the function and periodically calculate the covered hypervolume. In the end we calculate the mean hypervolume covered by the solution.

// Implementation of the MO-CMA-ES
#include <shark/Algorithms/DirectSearch/MOCMA.h>
#include <shark/ObjectiveFunctions/Benchmarks/ZDT1.h>
#include <shark/ObjectiveFunctions/Benchmarks/ZDT2.h>
#include <shark/ObjectiveFunctions/Benchmarks/ZDT3.h>
#include <shark/ObjectiveFunctions/Benchmarks/ZDT6.h>

using namespace shark;


The first question is how to measure the hypervolume. This is easy as shark comes with an algorithm for it. We only have to tell it how to extract the points from the solution object of the MO-CMA and choose a reference point. We will use the same reference as the one used for the analytic results for better comparison:

//functor returning the value vector of a solution object
struct PointExtractor{
template<class T>
RealVector const& operator()(T const& arg)const{
return arg.value;
}
};
template<class Solution>
double hypervolume( Solution const& solution){
// the reference point (11,11).
RealVector referencePoint(2,11);
//instance of the hypervolume calculator
HypervolumeCalculator hypervolume;
auto toPoints = [](typename Solution::const_reference point){return point.value;};
}


Next we set the experimental parameters:

std::size_t frontSize = 10; //number of points that approximate the front
std::size_t numDimensions = 10; //dimensions of the objective functions
std::size_t numTrials = 10; // how often the optimization is repeated
std::size_t recordingInterval = 20; //we want to record after some multiple of this
std::size_t numIterations = 20*recordingInterval; //number of iterations to perform


and create a vector of test functions:

//assortment of test functions
typedef boost::shared_ptr<MultiObjectiveFunction> Function;
std::vector<Function > functions;
functions.push_back(Function(new ZDT1(numDimensions)));
functions.push_back(Function(new ZDT2(numDimensions)));
functions.push_back(Function(new ZDT3(numDimensions)));
functions.push_back(Function(new ZDT6(numDimensions)));


The next block describes the experiment itself. Every function is optimized seperately. At the beginning of a trial, the MO-CMA is created and initialized, which will create an initial solution. The hypervolume of this solution is calculated before starting the optimization process which runs for the desired amount of iterations. The hypervolume values are accumulated in the matrix meanVolumes. After the optimization is finished, the accumulated sums are divided by the number of trials to get the mean:

RealMatrix meanVolumes(functions.size(), numIterations/recordingInterval+1,0.0);
for(std::size_t f = 0; f != functions.size(); ++f){
for(std::size_t trial = 0; trial != numTrials; ++trial){
//print progress
std::cout<<"\r" <<functions[f]->name() <<": "<<trial<<"/"<<numTrials<<std::flush;
//create and initialize the optimizer
MOCMA mocma;
mocma.mu() = frontSize;
functions[f]->init();
mocma.init( *functions[f] );

//record and hypervolume of initial solution
meanVolumes(f,0) += hypervolume(mocma.solution());

//optimize
for(std::size_t i = 1; i <= numIterations; ++i){
mocma.step(*functions[f]);
if(i % recordingInterval == 0){
meanVolumes(f,i / recordingInterval) += hypervolume(mocma.solution());
}
}
}
}
meanVolumes /= numTrials;


And we finally print the solution table together with a small legend:

std::cout<<"\r# Iteration ";
for(std::size_t f = 0; f != functions.size(); ++f)
std::cout<<functions[f]->name()<<" ";
std::cout<<"\n";

std::cout.precision( 7 );
for(std::size_t i = 0; i != meanVolumes.size2();++i){
std::cout<< i*recordingInterval<<" ";
for(std::size_t f = 0; f != functions.size(); ++f){
std::cout<<meanVolumes(f,i)<<" ";
}
std::cout<<"\n";
}


We can plot this with a simple gnuplot script which results in: