wfg — WFG Problem Collection

This module contains the test problems for multiobjective optimization published by the Walking Fish Group (WFG) [Huband2006]. The problems here are a reimplementation of the original C++ code at http://www.wfg.csse.uwa.edu.au/toolkit/ . They are all minimization problems.

class optproblems.wfg.WFG(num_objectives, num_variables, k, **kwargs)

The test problem collection WFG as defined in [Huband2006].

This class inherits from list and fills itself with the nine WFG problems with the chosen configuration. The arguments to the constructor are passed through to the problem classes.

References

[Huband2006]

(1, 2) Huband, S.; Hingston, P.; Barone, L.; While, L. (2006). A review of multiobjective test problems and a scalable test problem toolkit. IEEE Transactions on Evolutionary Computation, vol.10, no.5, pp. 477-506.

__init__(num_objectives, num_variables, k, **kwargs)

Constructor.

Parameters:

• num_objectives (int) – The number of objectives for the optimization problems.
• num_variables (int) – The number of decision variables of the problems.
• k (int) – The number of position related parameters, which are a part of the phenome. Thus it must hold k < num_variables. Furthermore, k must be a multiple of num_objectives - 1. Huband et al. recommend k = 4 for two objectives and k = 2 * (m - 1) for m objectives.
• kwargs – Arbitrary keyword arguments, passed through to the constructors of the single WFG problems.

class optproblems.wfg.WFGBaseProblem(objective_function, num_objectives, num_variables, k, **kwargs)

The base class for WFG problems.

get_optimal_solutions(max_number)

Return globally optimal or Pareto-optimal solutions.

This is an abstract method. Implementations must be deterministic. In the multi-objective case, the generated solutions should be evenly distributed over the whole Pareto-set.

m

The number of objective functions.

optimal_solution(k, l, pos_params)

Create one Pareto-optimal solution with given position parameters.

rand_optimal_solution(k, l, rand_generator=None)

Generate random position parameters and build optimal solution.

Test Problems

class optproblems.wfg.WFG1(num_objectives, num_variables, k, **kwargs)

The WFG1 problem.

class WFG1Shape

optimal_solution(k, l, pos_params)

Create one Pareto-optimal solution with given position parameters.

rand_optimal_solution(k, l, rand_generator=None)

Generate random position parameters and build optimal solution.

class optproblems.wfg.WFG2(num_objectives, num_variables, k, **kwargs)

The WFG2 problem.

class WFG2Shape

class optproblems.wfg.WFG3(num_objectives, num_variables, k, **kwargs)

The WFG3 problem.

class WFG3Shape

class optproblems.wfg.WFG4(num_objectives, num_variables, k, **kwargs)

The WFG4 problem.

class WFG4Shape

class optproblems.wfg.WFG5(num_objectives, num_variables, k, **kwargs)

The WFG5 problem.

class optproblems.wfg.WFG6(num_objectives, num_variables, k, **kwargs)

The WFG6 problem.

class optproblems.wfg.WFG7(num_objectives, num_variables, k, **kwargs)

The WFG7 problem.

class optproblems.wfg.WFG8(num_objectives, num_variables, k, **kwargs)

The WFG8 problem.

optimal_solution(k, l, pos_params)

Create one Pareto-optimal solution with given position parameters.

class optproblems.wfg.WFG9(num_objectives, num_variables, k, **kwargs)

The WFG9 problem.

optimal_solution(k, l, pos_params)

Create one Pareto-optimal solution with given position parameters.

Transformation Functions

optproblems.wfg.b_flat(y, a, b, c)

Transformation function.

optproblems.wfg.b_param(y, u, a, b, c)

Transformation function.

optproblems.wfg.b_poly(y, alpha)

Transformation function.

optproblems.wfg.r_nonsep(y, a)

Transformation function.

optproblems.wfg.r_sum(y, w)

Transformation function.

optproblems.wfg.s_decept(y, a, b, c)

Transformation function.

optproblems.wfg.s_linear(y, a)

Transformation function.

optproblems.wfg.s_multi(y, a, b, c)

Transformation function.

Shape Functions

optproblems.wfg.concave(x, m)

Shape function.

optproblems.wfg.convex(x, m)

Shape function.

optproblems.wfg.disc(x, a, alpha, beta)

Shape function.

optproblems.wfg.linear(x, m)

Shape function.

optproblems.wfg.mixed(x, a, alpha)

Shape function.

Helper Functions

optproblems.wfg.calculate_f(d, x, h, s)

Helper function.

optproblems.wfg.correct_to_01(a, epsilon=1e-10)

Sets values in [-epsilon, 0] to 0 and in [1, 1 + epsilon] to 1.

Assumption is that these deviations result from rounding errors.

optproblems.wfg.shape_args_ok(x, m)

Helper function.

optproblems.wfg.vector_in_01(x)

Returns True if all elements are in [0, 1].