Multi-objective benchmark problems#

namespace problems

Implementations of some benchmark functions that can be used to test the genetic algorithms. There are some benchmark problems implemented for every encoding type, and the real-encoded problems can also be used for the binary-encoded GAs.

All of the problems are implemented for maximization.

class Kursawe#

#include <problems/multi_objective.hpp>

class Kursawe : public gapp::problems::BenchmarkFunction<RealGene>#

Implementation of the Kursawe function for any number of variables, modified for maximization. It has 2 objectives, and the pareto front is made up of multiple disconnected segments.

Evaluated on the hypercube $x_{i} \in [- 5.0, 5.0]$ .

The approximate extreme points in the objective-space are:

ideal-point: (10 (n_{v a r s} - 1), 3.85 (n_{v a r s} - 1) + 4)

nadir-point: (7.25 (n_{v a r s} - 1), 0.0)

This benchmark function can be used for both the real- and binary-encoded GAs.

See also

Kursawe, F. “A variant of evolution strategies for vector optimization.” International conference on parallel problem solving from nature (1991): 193-197

Public Functions

explicit Kursawe(size_t num_vars = 3, size_t bits_per_var = 32)#

Create a Kursawe function.

Parameters:

num_vars – The number of variables. Must be at least 2.
bits_per_var – The number of bits representing a variable when used with the binary-encoded GA.

class ZDT1#

#include <problems/multi_objective.hpp>

class ZDT1 : public gapp::problems::BenchmarkFunction<RealGene>#

Implementation of the ZDT1 function for any number of variables, modified for maximization. This 2-objective benchmark function has a continuous, convex pareto front in the objective-space.

Evaluated on the hypercube $x_{i} \in [0.0, 1.0]$ .

The optimal solutions are $x_{1} \in [0.0, 1.0]$ , and $x_{r e s t} = 0.0$ .

The extreme points in the objective-space are:

ideal-point: (0.0, 0.0)

nadir-point: (- 1.0, - 1.0)

This benchmark function can be used for both the real- and binary-encoded GAs.

See also

Zitzler, E., Deb, K., and Thiele, L. “Comparison of multiobjective evolutionary algorithms: Empirical results.” Evolutionary computation 8, no. 2 (2000): 173-195.

Public Functions

explicit ZDT1(size_t num_vars = 30, size_t bits_per_var = 32)#

Create a ZDT1 function.

Parameters:

num_vars – The number of variables. Must be at least 2.
bits_per_var – The number of bits representing a variable when used with the binary-encoded GA.

class ZDT2#

#include <problems/multi_objective.hpp>

class ZDT2 : public gapp::problems::BenchmarkFunction<RealGene>#

Implementation of the ZDT2 function for any number of variables, modified for maximization. This 2-objective benchmark function has a continuous, non-convex pareto front in the objective-space.

Evaluated on the hypercube $x_{i} \in [0.0, 1.0]$ .

The optimal solutions are $x_{1} \in [0.0, 1.0]$ , and $x_{r e s t} = 0.0$ .

The extreme points in the objective-space are:

ideal-point: (0.0, 0.0)

nadir-point: (- 1.0, - 1.0)

This benchmark function can be used for both the real- and binary-encoded GAs.

See also

Zitzler, E., Deb, K., and Thiele, L. “Comparison of multiobjective evolutionary algorithms: Empirical results.” Evolutionary computation 8, no. 2 (2000): 173-195.

Public Functions

explicit ZDT2(size_t num_vars = 30, size_t bits_per_var = 32)#

Create a ZDT2 function.

Parameters:

num_vars – The number of variables. Must be at least 2.
bits_per_var – The number of bits representing a variable when used with the binary-encoded GA.

class ZDT3#

#include <problems/multi_objective.hpp>

class ZDT3 : public gapp::problems::BenchmarkFunction<RealGene>#

Implementation of the ZDT3 function for any number of variables, modified for maximization. This 2-objective benchmark function has a discontinuous pareto front made up of 5 disconnected segments in the objective-space.

Evaluated on the hypercube $x_{i} \in [0.0, 1.0]$ .

The optimal solutions are $x_{1} \in [0.0, 0.083] \cup [0.182, 0.258] \cup [0.409, 0.454] \cup [0.618 0.653] \cup [0.823, 0.852]$ , and $x_{r e s t} = 0.0$ .

The extreme points in the objective-space are:

ideal-point: (0.00, 0.8)

nadir-point: (- 0.85, - 1.0)

This benchmark function can be used for both the real- and binary-encoded GAs.

See also

Zitzler, E., Deb, K., and Thiele, L. “Comparison of multiobjective evolutionary algorithms: Empirical results.” Evolutionary computation 8, no. 2 (2000): 173-195.

Public Functions

explicit ZDT3(size_t num_vars = 30, size_t bits_per_var = 32)#

Create a ZDT3 function.

Parameters:

num_vars – The number of variables. Must be at least 2.
bits_per_var – The number of bits representing a variable when used with the binary-encoded GA.

class ZDT4#

#include <problems/multi_objective.hpp>

class ZDT4 : public gapp::problems::BenchmarkFunction<RealGene>#

Implementation of the ZDT4 function for any number of variables, modified for maximization. This 2-objective benchmark function is the most difficult problem of the ZDT suite. It has a large number of local pareto fronts in the objective space, which makes it easy for the GAs to get stuck along one of these local fronts.

Evaluated on $x_{1} \in [0.0, 1.0]$ , and $x_{r e s t} \in [- 5.0, 5.0]$ .

The optimal solutions are $x_{1} \in [0.0, 1.0]$ , and $x_{r e s t} = 0.0$ .

The extreme points in the objective-space are:

ideal-point: (0.0, 0.0)

nadir-point: (- 1.0, - 1.0)

This benchmark function can be used for both the real- and binary-encoded GAs.

See also

Zitzler, E., Deb, K., and Thiele, L. “Comparison of multiobjective evolutionary algorithms: Empirical results.” Evolutionary computation 8, no. 2 (2000): 173-195.

Public Functions

explicit ZDT4(size_t num_vars = 10, size_t bits_per_var = 32)#

Create a ZDT4 function.

Parameters:

num_vars – The number of variables. Must be at least 2.
bits_per_var – The number of bits representing a variable when used with the binary-encoded GA.

class ZDT5#

#include <problems/multi_objective.hpp>

class ZDT5 : public gapp::problems::BenchmarkFunction<BinaryGene>#

Implementation of the ZDT5 function for any number of variables, modified for maximization. In this 2-objective benchmark function, the variables are binary strings, not real numbers like they are in the rest of the functions of the ZDT suite.

The optimal solutions are $x_{1} = a n y t h i n g$ , and $x_{r e s t} = o n e s$ .

The extreme points in the objective-space are:

ideal-point: (- 1.0, - \frac{(n_{v a r s} - 1)}{31})

nadir-point: (- 31.0, - n_{v a r s} + 1)

This benchmark function can only be used with the binary-encoded GA, unlike the rest of the benchmark functions in the ZDT suite.

See also

Zitzler, E., Deb, K., and Thiele, L. “Comparison of multiobjective evolutionary algorithms: Empirical results.” Evolutionary computation 8, no. 2 (2000): 173-195.

Public Functions

explicit ZDT5(size_t num_vars = 11)#

Create a ZDT5 function.

Parameters:: num_vars – The number of variables. Must be at least 2.

class ZDT6#

#include <problems/multi_objective.hpp>

class ZDT6 : public gapp::problems::BenchmarkFunction<RealGene>#

Implementation of the ZDT6 function for any number of variables, modified for maximization. This 2-objective benchmark function has a non-convex pareto front in the objective space, along which the solutions are distributed non-uniformly.

Evaluated on the hypercube $x_{i} \in [0.0, 1.0]$ .

The optimal solutions are $x_{1} \in [0.0, 1.0]$ , and $x_{r e s t} = 0.0$ .

The extreme points in the objective-space are:

ideal-point: (0.0, 0.0)

nadir-point: (- 1.0, - 0.92)

This benchmark function can be used for both the real- and binary-encoded GAs.

See also

Zitzler, E., Deb, K., and Thiele, L. “Comparison of multiobjective evolutionary algorithms: Empirical results.” Evolutionary computation 8, no. 2 (2000): 173-195.

Public Functions

explicit ZDT6(size_t num_vars = 10, size_t bits_per_var = 32)#

Create a ZDT6 function.

Parameters:

num_vars – The number of variables. Must be at least 2.
bits_per_var – The number of bits representing a variable when used with the binary-encoded GA.

Multi-objective benchmark problems

Contents

Multi-objective benchmark problems#

class Kursawe#

class ZDT1#

class ZDT2#

class ZDT3#

class ZDT4#

class ZDT5#

class ZDT6#