problems.dynamic.gts¶

Classes¶

GTS1 ¶

GTS1(**kwargs)

Bases: GTS

GTS1 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = x_1 \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)(1 - (\frac{x_1}{g(\mathbf{x},t)})^{H(t)}) \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1)\), \(\mathbf{x}_{II,1} = (x_2, \cdots, x_{\lfloor\frac{D}{2}\rfloor})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 1}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \cos(0.5\pi t)\) and \(h_2(\mathbf{x}_I, t) = G(t) + x_1^{H(t)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1] \times [-1,1]^{\lfloor\frac{D}{2}\rfloor -1} \times [-1, 2]^{\lceil\frac{D}{2}\rceil}\).

Pareto set (PS)

GTS1 PS

Pareto front (PF)

GTS1 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=1, bounds=((0, 1), (-1, 1), (-1, 2)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    x = np.linspace(0, 1, n_pareto_points)
    f1 = x
    f2 = 1 - np.power(x, H_t(self.time))
    return np.array([f1, f2]).T

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = np.full(len(x_vec1), np.cos(0.5 * np.pi * self.time))
    x_vec3 = G_t(self.time) + np.power(x_vec1, H_t(self.time))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = np.cos(0.5 * np.pi * self.time)
    xj = G_t(self.time) + np.power(x[:, 0], H_t(self.time))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    f1 = x[:, 0]
    f2 = g * (1 - np.power(x[:, 0] / g, H_t(self.time)))

    out["F"] = np.column_stack([f1, f2])

GTS10 ¶

GTS10(**kwargs)

Bases: GTS

GTS10 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = g(\mathbf{x},t)\cos^2(0.5\pi x_1) \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)\cos^2(0.5\pi x_2) \\ f_3(\mathbf{x},t) = g(\mathbf{x},t)\sum_{j = 1}^{2}(\sin^2(0.5\pi x_j) + \sin(0.5\pi x_j)\cos^2(\lfloor6G(t)\rfloor \pi x_j)) \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1, x_2)\), \(\mathbf{x}_{II,1} = (x_3, \cdots, x_{\lfloor\frac{D}{2}\rfloor + 1})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 2}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \left\vert{G(t)}\right\vert\) and \(h_2(\mathbf{x}_I, t) = -0.5 + \frac{\left\vert{G(t)\sin(4\pi x_1)}\right\vert}{0.5(1+\left\vert{G(t)}\right\vert)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment,% the search space is \([0,1]^2 \times [0,1]^{\lfloor\frac{D}{2}\rfloor - 1} \times [-1, 1]^{\lceil\frac{D}{2}\rceil - 1}\).

Pareto set (PS)

GTS10 PS

Pareto front (PF)

GTS10 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=2, bounds=((0, 1), (0, 1), (-1, 1)), **kwargs)
    self.n_obj = 3

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    H = 20
    x1, x2 = np.meshgrid(np.linspace(0, 1, H), np.linspace(0, 1, H), indexing='xy')
    G = np.sin(0.5 * np.pi * self.time)
    p = np.floor(6 * G)

    f1 = np.cos(0.5 * np.pi * x1) ** 2
    f2 = np.cos(0.5 * np.pi * x2) ** 2
    f3 = np.sin(0.5 * np.pi * x1) ** 2 + np.sin(0.5 * np.pi * x1) * np.cos(p * np.pi * x1) ** 2 + np.sin(
        0.5 * np.pi * x2) ** 2 + np.sin(0.5 * np.pi * x2) * np.cos(p * np.pi * x2) ** 2

    return get_PF(np.array([f1, f2, f3]), True)

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = np.full(len(x_vec1), np.abs(G_t(self.time)))
    x_vec3 = -0.5 + np.abs(G_t(self.time) * np.sin(4 * np.pi * x_vec1)) / (0.5 * (1 + np.abs(G_t(self.time))))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = np.abs(G_t(self.time))
    xj = -0.5 + np.abs(G_t(self.time) * np.sin(4 * np.pi * x[:, 1])) / (0.5 * (1 + np.abs(G_t(self.time))))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    f1 = g * np.cos(0.5 * np.pi * x[:, 0]) ** 2
    f2 = g * np.cos(0.5 * np.pi * x[:, 1]) ** 2

    _sin1 = np.sin(0.5 * np.pi * x[:, 0])
    _sin2 = np.sin(0.5 * np.pi * x[:, 1])
    _cos1 = np.cos(p_t(self.time) * np.pi * x[:, 0])
    _cos2 = np.cos(p_t(self.time) * np.pi * x[:, 1])
    f3 = g * (_sin1 ** 2 + _sin1 * _cos1 ** 2 + _sin2 ** 2 + _sin2 * _cos2 ** 2)

    out["F"] = np.column_stack([f1, f2, f3])

GTS10_2 ¶

GTS10_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS10_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS10_3 ¶

GTS10_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS10_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS11 ¶

GTS11(**kwargs)

Bases: GTS

GTS11 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = g(\mathbf{x},t)(1.05 - y + 0.05\sin(6\pi y)) \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)(1.05 - x_2 + 0.05\sin(6\pi x_2))(y + 0.05\sin(6\pi y)) \\ f_3(\mathbf{x},t) = g(\mathbf{x},t)(x_2 + 0.05\sin(6\pi x_2))(y + 0.05\sin(6\pi y)) \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1, x_2)\), \(\mathbf{x}_{II,1} = (x_3, \cdots, x_{\lfloor\frac{D}{2}\rfloor + 1})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 2}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \left\vert{G(t)}\right\vert\) and \(h_2(\mathbf{x}_I, t) = G(t) + x_1^{H(t)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1]^2 \times [0,1]^{\lfloor\frac{D}{2}\rfloor - 1} \times [-1, 2]^{\lceil\frac{D}{2}\rceil - 1}\).

Pareto set (PS)

GTS11 PS

Pareto front (PF)

GTS11 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=2, bounds=((0, 1), (0, 1), (-1, 2)), **kwargs)
    self.n_obj = 3

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    H = 20
    x1, x2 = np.meshgrid(np.linspace(0, 1, H), np.linspace(0, 1, H), indexing='xy')

    y = y_t(x1, self.time)
    f1 = 1.05 - y + 0.05 * np.sin(6 * np.pi * y)
    f2 = np.multiply(1.05 - x2 + 0.05 * np.sin(6 * np.pi * x2), y + 0.05 * np.sin(6 * np.pi * y))
    f3 = np.multiply(x2 + 0.05 * np.sin(6 * np.pi * x2), y + 0.05 * np.sin(6 * np.pi * y))

    return get_PF(np.array([f1, f2, f3]), False)

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = np.full(len(x_vec1), np.abs(G_t(self.time)))
    x_vec3 = G_t(self.time) + np.power(x_vec1, H_t(self.time))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = np.abs(G_t(self.time))
    xj = G_t(self.time) + np.power(x[:, 0], H_t(self.time))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    y = y_t(x[:, 0], self.time)
    f1 = g * (1.05 - y + 0.05 * np.sin(6 * np.pi * y))
    f2 = g * (1.05 - x[:, 1] + 0.05 * np.sin(6 * np.pi * x[:, 1])) * (y + 0.05 * np.sin(6 * np.pi * y))
    f3 = g * (x[:, 1] + 0.05 * np.sin(6 * np.pi * x[:, 1])) * (y + 0.05 * np.sin(6 * np.pi * y))

    out["F"] = np.column_stack([f1, f2, f3])

GTS11_2 ¶

GTS11_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS11_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS11_3 ¶

GTS11_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS11_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS1_2 ¶

GTS1_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS1_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS1_3 ¶

GTS1_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS1_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS2 ¶

GTS2(**kwargs)

Bases: GTS

GTS2 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = 0.5x_1+x_2 \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)(2.8 - (\frac{0.5x_1+x_2}{g(\mathbf{x},t)})^{H(t)}) \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1, x_2)\), \(\mathbf{x}_{II,1} = (x_3, \cdots, x_{\lfloor\frac{D}{2}\rfloor + 1})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 2}, \cdots, x_D)\), \(c = \cot(3\pi t^2), \text{when~} t^2 \neq \frac{n}{3}, n \in \mathbb{Z}, c = 1e-32, \text{otherwise}\), \(h_1(\mathbf{x}_I, t) = \frac{1}{\pi}\left\vert{\arctan(c)}\right\vert\) and \(h_2(\mathbf{x}_I, t) = G(t) + x_1^{H(t)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1]^2 \times [0,1]^{\lfloor\frac{D}{2}\rfloor -1} \times [-1, 2]^{\lceil\frac{D}{2}\rceil-1}\).

Pareto set (PS)

GTS2 PS

Pareto front (PF)

GTS2 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=2, bounds=((0, 1), (0, 1), (-1, 2)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    H = 20
    x1, x2 = np.meshgrid(np.linspace(0, 1, H), np.linspace(0, 1, H), indexing='xy')

    f1 = 0.5 * x1 + x2
    f2 = 2.8 - np.power(f1, H_t(self.time))
    return get_PF(np.array([f1, f2]), False)

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    t = 3 * np.pi * self.time ** 2
    cot = np.cos(t) / (np.sin(t) + (np.sin(t) == 0) * 1e-32)

    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = np.full(len(x_vec1), (1 / np.pi) * np.abs(np.arctan(cot)))
    x_vec3 = G_t(self.time) + np.power(x_vec1, H_t(self.time))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    t = 3 * np.pi * self.time ** 2
    cot = np.cos(t) / (np.sin(t) + (np.sin(t) == 0) * 1e-32)

    xi = (1 / np.pi) * np.abs(np.arctan(cot))
    xj = G_t(self.time) + np.power(x[:, 0], H_t(self.time))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    f1 = 0.5 * x[:, 0] + x[:, 1]
    f2 = g * (2.8 - np.power(f1 / g, H_t(self.time)))

    out["F"] = np.column_stack([f1, f2])

GTS2_2 ¶

GTS2_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS2_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS2_3 ¶

GTS2_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS2_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS3 ¶

GTS3(**kwargs)

Bases: GTS

GTS3 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = g(x)(x_1 + 0.1\sin(3\pi x_1))^{\beta_t} \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)(1 - x_1 + 0.1\sin(3\pi x_1))^{\beta_t} \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1)\), \(\mathbf{x}_{II,1} = (x_2, \cdots, x_{\lfloor\frac{D}{2}\rfloor})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 1}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \frac{G(t)\sin(4\pi x_1)}{1 + \left\vert{G(t)}\right\vert}\) and \(h_2(\mathbf{x}_I, t) = G(t) + x_1^{H(t)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1] \times [-1,1]^{\lfloor\frac{D}{2}\rfloor - 1} \times [-1, 2]^{\lceil\frac{D}{2}\rceil}\).

Pareto set (PS)

GTS3 PS

Pareto front (PF)

GTS3 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=1, bounds=((0, 1), (-1, 1), (-1, 2)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    x = np.linspace(0, 1, n_pareto_points)
    f1 = np.power(x + 0.1 * np.sin(3 * np.pi * x), beta_t(self.time))
    f2 = np.power(1 - x + 0.1 * np.sin(3 * np.pi * x), beta_t(self.time))

    return get_PF(np.array([f1, f2]), True)

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = (G_t(self.time) * np.sin(4 * np.pi * x_vec1)) / (1 + np.abs(G_t(self.time)))
    x_vec3 = G_t(self.time) + np.power(x_vec1, H_t(self.time))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = (G_t(self.time) * np.sin(4 * np.pi * x[:, 0])) / (1 + np.abs(G_t(self.time)))
    xj = G_t(self.time) + np.power(x[:, 0], H_t(self.time))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    f1 = g * np.power(x[:, 0] + 0.1 * np.sin(3 * np.pi * x[:, 0]), beta_t(self.time))
    f2 = g * np.power(1 - x[:, 0] + 0.1 * np.sin(3 * np.pi * x[:, 0]), beta_t(self.time))

    out["F"] = np.column_stack([f1, f2])

GTS3_2 ¶

GTS3_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS3_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS3_3 ¶

GTS3_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS3_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS4 ¶

GTS4(**kwargs)

Bases: GTS

GTS4 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = g(\mathbf{x},t)\frac{1 + t}{x_1 + 3} \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)\frac{x_1 + 3}{1 + t} \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & - 0.5 + 0.25\sin(0.3\pi t) \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1)\), \(\mathbf{x}_{II,1} = (x_2, \cdots, x_{\lfloor\frac{D}{2}\rfloor})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 1}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \left\vert{G(t)}\right\vert\) and \(h_2(\mathbf{x}_I, t) = \frac{G(t)\sin(4\pi x_1)}{1 + \left\vert{G(t)}\right\vert}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1] \times [0,1]^{\lfloor\frac{D}{2}\rfloor -1} \times [-1, 1]^{\lceil\frac{D}{2}\rceil}\).

Pareto set (PS)

GTS4 PS

Pareto front (PF)

GTS4 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=1, bounds=((0, 1), (0, 1), (-1, 1)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    x = np.linspace(0, 1, n_pareto_points)
    g = 1 + (-0.5 + 0.25 * np.sin(0.3 * np.pi * self.time))

    f1 = g * ((1 + self.time) / (x + 3))
    f2 = g * ((x + 3) / (1 + self.time))

    return np.array([f1, f2]).T

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = np.full(len(x_vec1), np.abs(G_t(self.time)))
    x_vec3 = (G_t(self.time) * np.sin(4 * np.pi * x_vec1)) / (1 + np.abs(G_t(self.time)))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = np.abs(G_t(self.time))
    xj = (G_t(self.time) * np.sin(4 * np.pi * x[:, 0])) / (1 + np.abs(G_t(self.time)))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    g += (-0.5 + 0.25 * np.sin(0.3 * np.pi * self.time))

    f1 = g * ((1 + self.time) / (x[:, 0] + 3))
    f2 = g * ((x[:, 0] + 3) / (1 + self.time))

    out["F"] = np.column_stack([f1, f2])

GTS4_2 ¶

GTS4_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS4_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS4_3 ¶

GTS4_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS4_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS5 ¶

GTS5(**kwargs)

Bases: GTS

GTS5 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = g(\mathbf{x},t)((0.5x_1+x_2)) + 0.02\sin(\omega_t\pi (0.5x_1+x_2))) \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)(1.6 - (0.5x_1+x_2) + 0.02\sin(\omega_t\pi (0.5x_1+x_2))) \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + 0.5 + 0.5G(t) \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1, x_2)\), \(\mathbf{x}_{II,1} = (x_3, \cdots, x_{\lfloor\frac{D}{2}\rfloor + 1})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 2}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \cos(0.5\pi t)\) and \(h_2(\mathbf{x}_I, t) = G(t) + x_1^{H(t)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1]^2 \times [-1,1]^{\lfloor\frac{D}{2}\rfloor - 1} \times [-1, 2]^{\lceil\frac{D}{2}\rceil-1}\).

Pareto set (PS)

GTS5 PS

Pareto front (PF)

GTS5 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=2, bounds=((0, 1), (-1, 1), (-1, 2)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    H = 20
    x1, x2 = np.meshgrid(np.linspace(0, 1, H), np.linspace(0, 1, H), indexing='xy')
    g = 1 + 0.5 + 0.5 * G_t(self.time)

    x12 = 0.5 * x1 + x2
    _sin = 0.02 * np.sin(omega_t(self.time) * np.pi * x12)
    f1 = g * (x12 + _sin)
    f2 = g * (1.6 - x12 + _sin)

    return get_PF(np.array([f1, f2]), True)

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = np.full(len(x_vec1), np.cos(0.5 * np.pi * self.time))
    x_vec3 = G_t(self.time) + np.power(x_vec1, H_t(self.time))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = np.cos(0.5 * np.pi * self.time)
    xj = G_t(self.time) + np.power(x[:, 0], H_t(self.time))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    g += 0.5 + 0.5 * G_t(self.time)

    x12 = 0.5 * x[:, 0] + x[:, 1]
    _sin = 0.02 * np.sin(omega_t(self.time) * np.pi * x12)
    f1 = g * (x12 + _sin)
    f2 = g * (1.6 - x12 + _sin)

    out["F"] = np.column_stack([f1, f2])

GTS5_2 ¶

GTS5_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS5_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS5_3 ¶

GTS5_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS5_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS6 ¶

GTS6(**kwargs)

Bases: GTS

GTS6 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = x_1 \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)(1 - (\frac{x_1}{g(\mathbf{x},t)})^{H(t)}) \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1)\), \(\mathbf{x}_{II,1} = (x_2, \cdots, x_{\lfloor\frac{D}{2}\rfloor})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 1}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \cos(0.5\pi t)\) and \(h_2(\mathbf{x}_I, t) = G(t) + x_1^{H(t)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1] \times [-1,1]^{\lfloor\frac{D}{2}\rfloor -1} \times [-1, 2]^{\lceil\frac{D}{2}\rceil}\).

Pareto set (PS)

GTS6 PS

Pareto front (PF)

GTS6 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=1, bounds=((0, 1), (-1, 1), (-1, 2)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    f1 = np.linspace(0, 1, n_pareto_points)
    f2 = 1 - np.power(f1, H_t(self.time))
    return np.array([f1, f2]).T

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = np.full(len(x_vec1), np.cos(0.5 * np.pi * self.time))
    x_vec3 = G_t(self.time) + np.power(x_vec1, H_t(self.time))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = self.time_linkage * np.cos(0.5 * np.pi * self.time)
    xj = self.time_linkage * (G_t(self.time) + np.power(x[:, 0], H_t(self.time)))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    f1 = x[:, 0]
    f2 = g * (1 - np.power(f1 / g, H_t(self.time)))

    out["F"] = np.column_stack([f1, f2])

GTS6_2 ¶

GTS6_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS6_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS6_3 ¶

GTS6_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS6_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS7 ¶

GTS7(**kwargs)

Bases: GTS

GTS7 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = g(\mathbf{x},t)\left\vert{x_1-a_t}\right\vert^{H(t)} \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)\left\vert{x_1-a_t-b_t}\right\vert^{H(t)} \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1)\), \(\mathbf{x}_{II,1} = (x_2, \cdots, x_{\lfloor\frac{D}{2}\rfloor})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 1}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \cos(0.5\pi t)\) and \(h_2(\mathbf{x}_I, t) = \frac{1}{1 + e^{\alpha_t(x_1 - 0.5)}}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([-1,2.5] \times [-1,1]^{\lfloor\frac{D}{2}\rfloor - 1} \times [0, 1]^{\lceil\frac{D}{2}\rceil}\).

Pareto set (PS)

GTS7 PS

Pareto front (PF)

GTS7 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=1, bounds=((-1, 2.5), (-1, 1), (0, 1)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    x = np.linspace(a_t(self.time), a_t(self.time) + b_t(self.time), n_pareto_points)

    f1 = np.power(np.abs(x - a_t(self.time)), H_t(self.time))
    f2 = np.power(np.abs(x - a_t(self.time) - b_t(self.time)), H_t(self.time))

    return np.array([f1, f2]).T

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(a_t(self.time), a_t(self.time) + b_t(self.time), n_pareto_points)
    x_vec2 = np.full(len(x_vec1), np.cos(0.5 * np.pi * self.time))
    x_vec3 = 1 / (1 + np.exp(alpha_t(self.time) * (x_vec1 - 0.5)))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = self.time_linkage * np.cos(0.5 * np.pi * self.time)
    xj = self.time_linkage * 1 / (1 + np.exp(alpha_t(self.time) * (x[:, 0] - 0.5)))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    f1 = g * np.power(np.abs(x[:, 0] - a_t(self.time)), H_t(self.time))
    f2 = g * np.power(np.abs(x[:, 0] - a_t(self.time) - b_t(self.time)), H_t(self.time))

    out["F"] = np.column_stack([f1, f2])

GTS7_2 ¶

GTS7_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS7_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS7_3 ¶

GTS7_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS7_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS8 ¶

GTS8(**kwargs)

Bases: GTS

GTS8 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = (0.5x_1+x_2) \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)(2.8 - (\frac{(0.5x_1+x_2)}{g(\mathbf{x},t)})^{H(t)}) \\ \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + 0.25\left\vert{\cos(0.3 \pi t)}\right\vert \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1, x_2)\), \(\mathbf{x}_{II,1} = (x_3, \cdots, x_{\lfloor\frac{D}{2}\rfloor + 1})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 2}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \frac{1}{1 + e^{\alpha _t(x_1 - 0.5)}}\) and \(h_2(\mathbf{x}_I, t) = G(t) + x_1^{H(t)}\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1]^2 \times [0,1]^{\lfloor\frac{D}{2}\rfloor -1} \times [-1, 2]^{\lceil\frac{D}{2}\rceil-1}\).

Pareto set (PS)

GTS8 PS

Pareto front (PF)

GTS8 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=2, bounds=((0, 1), (0, 1), (-1, 2)), **kwargs)

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    H = 20
    x1, x2 = np.meshgrid(np.linspace(0, 1, H), np.linspace(0, 1, H), indexing='xy')

    f1 = 0.5 * x1 + x2
    g = 1 + 0.25 * np.abs(np.cos(0.3 * np.pi * self.time))
    f2 = g * (2.8 - np.power(f1 / g, H_t(self.time)))
    return get_PF(np.array([f1, f2]), False)

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = 1 / (1 + np.exp(alpha_t(self.time) * (x_vec1 - 0.5)))
    x_vec3 = G_t(self.time) + np.power(x_vec1, H_t(self.time))
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = self.time_linkage * 1 / (1 + np.exp(alpha_t(self.time) * (x[:, 0] - 0.5)))
    xj = self.time_linkage * (G_t(self.time) + np.power(x[:, 0], H_t(self.time)))

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    g += 0.25 * np.abs(np.cos(0.3 * np.pi * self.time))

    f1 = 0.5 * x[:, 0] + x[:, 1]
    f2 = g * (2.8 - np.power(f1 / g, H_t(self.time)))

    out["F"] = np.column_stack([f1, f2])

GTS8_2 ¶

GTS8_2(matrix_case='two', **kwargs)

Bases: GTS1

GTS8_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="two", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS8_3 ¶

GTS8_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS8_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)

GTS9 ¶

GTS9(**kwargs)

Bases: GTS

GTS9 test problem.

Inherits all parameters from parent class GTS.

Attributes:

Name	Type	Description
`name`	`str`	Problem name, default is 'GTS1'
`n_var`	`int`	Number of decision variables
`n_obj`	`int`	Number of objective functions
`time_linkage`	`bool`	Whether the problem has time linkage

Notes

Mathematical Formulation:

\[\begin{equation} \text{min} \begin{cases} f_1(\mathbf{x},t) = g(\mathbf{x},t)\cos(0.5\pi x_1)\cos(0.5\pi x_2) \\ f_2(\mathbf{x},t) = g(\mathbf{x},t)\cos(0.5\pi x_1)\sin(0.5\pi x_2) \\ f_3(\mathbf{x},t) = g(\mathbf{x},t)\sin(0.5\pi x_1) \end{cases} \end{equation}\]

with

\[\begin{equation*} \begin{split} g(\mathbf{x},t) = 1 & + \Bigl(\bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,1}(t) \bigl(\mathbf{x}_{II,1} - h_1(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \Bigl(\bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)^T \mathbf{R}_{II,2}(t) \bigl(\mathbf{x}_{II,2} - h_2(\mathbf{x}_I)\bigr)\Bigr)^{\frac{1}{p}} \\ & + \left\vert{\cos(0.27\pi t)}\right\vert \end{split} \end{equation*}\]

where \(p \geq 1\), \(\mathbf{x}_I = (x_1, x_2)\), \(\mathbf{x}_{II,1} = (x_3, \cdots, x_{\lfloor\frac{D}{2}\rfloor + 1})\) and \(\mathbf{x}_{II,2} = (x_{\lfloor\frac{D}{2}\rfloor + 2}, \cdots, x_D)\), \(h_1(\mathbf{x}_I, t) = \frac{1}{1+e^{\alpha_t(x_1 - 0.5)}}\) and \(h_2(\mathbf{x}_I, t) = \sin(tx_1)\), \(\mathbf{R}_{II,1}(t)\) and \(\mathbf{R}_{II,2}(t)\) are symmetric positive semidefinite matrices in the \(t\)-th environment, the search space is \([0,1]^2 \times [0,1]^{\lfloor\frac{D}{2}\rfloor - 1} \times [-1, 1]^{\lceil\frac{D}{2}\rceil - 1}\).

Pareto set (PS)

GTS9 PS

Pareto front (PF)

GTS9 PF

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, **kwargs):
    super().__init__(part_idx=2, bounds=((0, 1), (0, 1), (-1, 1)), **kwargs)
    self.n_obj = 3

Functions¶

_calc_pareto_front ¶

_calc_pareto_front(*args, n_pareto_points=100, **kwargs)

Pareto front.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_front(self, *args, n_pareto_points=100, **kwargs):
    """Pareto front."""
    H = 20
    x1, x2 = np.meshgrid(np.linspace(0, 1, H), np.linspace(0, 1, H), indexing='xy')

    g = 1
    g += np.abs(np.cos(0.27 * np.pi * self.time))

    f1 = np.multiply(np.multiply(g, np.cos(0.5 * np.pi * x2)), np.cos(0.5 * np.pi * x1))
    f2 = np.multiply(np.multiply(g, np.sin(0.5 * np.pi * x2)), np.cos(0.5 * np.pi * x1))
    f3 = np.multiply(g, np.sin(0.5 * np.pi * x1))

    return get_PF(np.array([f1, f2, f3]), True)

_calc_pareto_set ¶

_calc_pareto_set(*args, n_pareto_points=100, **kwargs)

Pareto set.

Source code in pydmoo/problems/dynamic/gts.py

def _calc_pareto_set(self, *args, n_pareto_points=100, **kwargs):
    """Pareto set."""
    x_vec1 = np.linspace(0, 1, n_pareto_points)
    x_vec2 = 1 / (1 + np.exp(alpha_t(self.time) * (x_vec1 - 0.5)))
    x_vec3 = np.sin(self.time * x_vec1)
    X, Y, Z = x_vec1, x_vec2, x_vec3
    return np.array([X.flatten(order='F'), Y.flatten(order='F'), Z.flatten(order='F')]).T

_evaluate ¶

_evaluate(x, out, *args, **kwargs)

Evaluate.

Source code in pydmoo/problems/dynamic/gts.py

def _evaluate(self, x, out, *args, **kwargs):
    """Evaluate."""
    xi = 1 / (1 + np.exp(alpha_t(self.time) * (x[:, 0] - 0.5)))
    xj = np.sin(self.time * x[:, 0])

    x2, x3 = (x[:, self.sub_vec_2] - xi.reshape(-1, 1)), (x[:, self.sub_vec_3] - xj.reshape(-1, 1))

    # quadratic form
    diag_x2_R2_x2T = np.einsum('ij,jk,ik->i', x2, self.matrix_2, x2)
    diag_x3_R3_x3T = np.einsum('ij,jk,ik->i', x3, self.matrix_3, x3)
    g = 1 + np.power(diag_x2_R2_x2T, self.p) + np.power(diag_x3_R3_x3T, self.p)

    g += np.abs(np.cos(0.27 * np.pi * self.time))

    f1 = g * np.cos(0.5 * np.pi * x[:, 1]) * np.cos(0.5 * np.pi * x[:, 0])
    f2 = g * np.sin(0.5 * np.pi * x[:, 1]) * np.cos(0.5 * np.pi * x[:, 0])
    f3 = g * np.sin(0.5 * np.pi * x[:, 0])

    out["F"] = np.column_stack([f1, f2, f3])

GTS9_2 ¶

GTS9_2(matix_case='two', **kwargs)

Bases: GTS1

GTS9_2 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matix_case="two", **kwargs):
    super().__init__(matrix_case=matix_case, **kwargs)

GTS9_3 ¶

GTS9_3(matrix_case='three', **kwargs)

Bases: GTS1

GTS9_3 test problem.

Source code in pydmoo/problems/dynamic/gts.py

def __init__(self, matrix_case="three", **kwargs):
    super().__init__(matrix_case=matrix_case, **kwargs)