import numpy as np
[docs]class Rectangles(object):
def __init__(self, n_dim, dtype):
"""
Initilize a set of hyper-rectangle.
:param n_dim: dimension of rectangles
"""
self.n_dim = n_dim
self.lb = np.zeros((0, self.n_dim), dtype=dtype)
self.ub = np.zeros((0, self.n_dim), dtype=dtype)
[docs] def add(self, lb, ub):
"""
Add new rectangles.
:param lb: lower bounds of rectangles
:param ub: upper bounds of rectangles
"""
self.lb = np.r_[self.lb, lb]
self.ub = np.r_[self.ub, ub]
[docs]def dominate(t1, t2):
"""domination rule for maximization problem"""
return np.all(t1 >= t2) and np.any(t1 > t2)
[docs]class Pareto(object):
def __init__(self, num_objectives, dom_rule=None):
self.num_objectives = num_objectives
self.front = np.zeros((0, self.num_objectives))
self.front_num = np.zeros(0, dtype=int)
self.num_compared = 0
self.dom_rule = dom_rule
self.front_updated = False
if self.dom_rule is None:
self.dom_rule = dominate
self.cells = Rectangles(num_objectives, int)
self.reference_min = None
self.reference_max = None
[docs] def update_front(self, t):
"""
Update the non-dominated set of points.
Pareto set is sorted on the first objective in ascending order.
"""
t = np.array(t)
if t.ndim == 1:
tt = [t]
else:
tt = t
front_updated = False
for k in range(len(tt)):
point = tt[k]
is_front = True
for i in range(len(self.front)):
if self.dom_rule(self.front[i], point):
is_front = False
break
if is_front:
front_updated = True
dom_filter = np.full(len(self.front), True, dtype=bool)
for i in range(len(self.front)):
if self.dom_rule(point, self.front[i]):
dom_filter[i] = False
self.front = np.r_[self.front[dom_filter], point[np.newaxis, :]]
self.front_num = np.r_[self.front_num[dom_filter], self.num_compared]
self.num_compared += 1
if front_updated:
sorted_idx = self.front[:, 0].argsort()
self.front = self.front[sorted_idx, :]
self.front_num = self.front_num[sorted_idx]
self.divide_non_dominated_region()
self.front_updated = front_updated
[docs] def export_front(self):
return self.front, self.front_num
[docs] def set_reference_min(self, reference_min=None):
if reference_min is None:
# estimate reference min point
front_min = np.min(self.front, axis=0, keepdims=True)
w = np.max(self.front, axis=0, keepdims=True) - front_min
reference_min = front_min - w * 2 / self.front.shape[0]
self.reference_min = reference_min
[docs] def set_reference_max(self, reference_max=None):
if reference_max is None:
# estimate reference max point
front_max = np.max(self.front, axis=0, keepdims=True)
w = front_max - np.min(self.front, axis=0, keepdims=True)
reference_max = front_max + w * 100
self.reference_max = reference_max
[docs] def volume_in_dominance(self, ref_min, ref_max, dominance_ratio=False):
ref_min = np.array(ref_min)
ref_max = np.array(ref_max)
v_all = np.prod(ref_max - ref_min)
front = np.r_[[ref_min], self.front, [ref_max]]
ax = np.arange(self.num_objectives)
lb = front[self.cells.lb, ax]
ub = front[self.cells.ub, ax]
v_non_dom = np.sum(np.prod(ub - lb, axis=1))
if dominance_ratio:
return (v_all - v_non_dom) / v_all
else:
return v_all - v_non_dom
[docs] def divide_non_dominated_region(self, force_binary_search=False):
# clear rectangles
self.cells = Rectangles(self.num_objectives, int)
if self.num_objectives == 2 and not force_binary_search:
self.__divide_2d()
else:
self.__divide_using_binary_search()
def __divide_2d(self):
"""
Divide non-dominated region into vertical rectangles for the case of 2-objectives.
Assumes that Pareto set has been sorted on the first objective in ascending order.
Notes:
In 2-dimensional cases, the second objective has be sorted in decending order.
"""
n_cells = self.front.shape[0] + 1
lb_idx = [[i, (i + 1) % n_cells] for i in range(n_cells)]
ub_idx = [[i + 1, n_cells] for i in range(n_cells)]
self.cells.add(lb_idx, ub_idx)
def __included_in_non_dom_region(self, p):
return np.all([np.any(pf <= p) for pf in self.front])
def __divide_using_binary_search(self):
front = np.r_[
np.full((1, self.num_objectives), -np.inf),
self.front,
np.full((1, self.num_objectives), np.inf),
]
# Pareto front indices when sorted on each dimension's front value in ascending order.
# (indices start from 1)
# Index 0 means anti-ideal value, index `self.front.shape[0] + 1` means ideal point.
front_idx = np.r_[
np.zeros((1, self.num_objectives), dtype=int),
np.argsort(self.front, axis=0) + 1,
np.full((1, self.num_objectives), self.front.shape[0] + 1, dtype=int),
]
rect_candidates = [[np.copy(front_idx[0]), np.copy(front_idx[-1])]]
while rect_candidates:
rect = rect_candidates.pop()
lb_idx = [front_idx[rect[0][d], d] for d in range(self.num_objectives)]
ub_idx = [front_idx[rect[1][d], d] for d in range(self.num_objectives)]
lb = [front[lb_idx[d], d] for d in range(self.num_objectives)]
ub = [front[ub_idx[d], d] for d in range(self.num_objectives)]
if self.__included_in_non_dom_region(lb):
self.cells.add([lb_idx], [ub_idx])
elif self.__included_in_non_dom_region(ub):
rect_sizes = rect[1] - rect[0]
# divide rectangle by the dimension with largest size
if np.any(rect_sizes > 1):
div_dim = np.argmax(rect_sizes)
div_point = rect[0][div_dim] + int(round(rect_sizes[div_dim] / 2.0))
# add divided left rectangle
left_ub_idx = np.copy(rect[1])
left_ub_idx[div_dim] = div_point
rect_candidates.append([np.copy(rect[0]), left_ub_idx])
# add divided right rectangle
right_lb_idx = np.copy(rect[0])
right_lb_idx[div_dim] = div_point
rect_candidates.append([right_lb_idx, np.copy(rect[1])])