def _advance(self, **kwargs):
# all the elements in the interval
a, c, d, b = self.pop
# the golden ratio (precomputed constant)
R = self.R
# if the left solution is better than the right
if c.F[0] < d.F[0]:
# make the right to be the new right bound and the left becomes the right
a, b = a, d
d = c
# create a new left individual and evaluate
c = Individual(X=b.X - R * (b.X - a.X))
self.evaluator.eval(self.problem, c, algorithm=self)
self.infills = c
# if the right solution is better than the left
else:
# make the left to be the new left bound and the right becomes the left
a, b = c, b
c = d
# create a new right individual and evaluate
d = Individual(X=a.X + R * (b.X - a.X))
self.evaluator.eval(self.problem, d, algorithm=self)
self.infills = d
# update the population with all the four individuals
self.pop = Population.create(a, c, d, b)
def zoom(alpha_low, alpha_high, max_iter=100):
while True:
_alpha = (alpha_high.get("alpha") + alpha_low.get("alpha")) / 2
_point = Individual(X=_alpha)
evaluator.eval(problem, _point, evaluate_values_of=["F", "CV"])
if _point.F[
0] > sol_F + self.c1 * _alpha * sol_dF @ direction or _point.F[
0] > alpha_low.F[0]:
alpha_high = _point
else:
evaluator.eval(problem, _point, evaluate_values_of=["dF"])
point_dF = _point.get("dF")[0]
if np.abs(point_dF
@ direction) <= -self.c2 * sol_dF @ direction:
return _point
if (point_dF @ direction) * (alpha_high.get("alpha") -
alpha_low.get("alpha")) >= 0:
alpha_high = alpha_low
alpha_low = _point
def _initialize_infill(self):
# x could be a vector or an individual
if isinstance(self.point, np.ndarray):
self.point = Individual(X=self.point)
# make sure it is evaluated - if not yet also get the gradient
if self.point.get("F") is None:
self.evaluator.eval(self.problem, self.point, algorithm=self)
self.infill = self.point
class LineSearch(Algorithm):
def __init__(self, **kwargs):
super().__init__(**kwargs)
self.point, self.direction = None, None
def setup(self, problem, point=None, direction=None, **kwargs):
super().setup(problem, **kwargs)
msg = "Only problems with one objective and no constraints can be solved using a line search!"
assert not problem.has_constraints() and problem.n_obj == 1, msg
assert point is not None, "You have to define a starting point for the algorithm"
self.point = point
assert direction is not None, "You have to define a direction point for the algorithm"
self.direction = direction
return self
def _initialize_infill(self):
# x could be a vector or an individual
if isinstance(self.point, np.ndarray):
self.point = Individual(X=self.point)
# make sure it is evaluated - if not yet also get the gradient
if self.point.get("F") is None:
self.evaluator.eval(self.problem, self.point, algorithm=self)
self.infill = self.point
def _initialize_infill(self):
super()._initialize_infill()
a, b = self.a, self.b
# the golden ratio (precomputed constant)
R = self.R
# create the left and right in the interval itself
c, d = Individual(X=b.X - R * (b.X - a.X)), Individual(X=a.X + R *
(b.X - a.X))
# create a population with all four individuals
pop = Population.create(a, c, d, b)
self.pop, self.infills = pop, pop
return pop
def _pattern_move(self, _current, _direction):
# calculate the new X and repair out of bounds if necessary
X = _current.X + self.step_size * _direction
set_to_bounds_if_outside_by_problem(self.problem, X)
# create the new center individual and evaluate it
trial = Individual(X=X)
self.evaluator.eval(self.problem, trial, algorithm=self)
return trial
def _initialize_infill(self):
super()._initialize_infill()
a, b = self.a, self.b
# set c to be directly in the middle between the two brackets
c = Individual(X=(b.X - a.X) / 2)
# create a population with all three individuals
pop = Population.create(a, b, c)
return pop
def _infill(self):
# the offspring population to finally evaluate and attach to the population
infills = Population()
# find the potential optimal solution in the current population
potential_optimal = self._potential_optimal()
# for each of those solutions execute the division move
for current in potential_optimal:
# find the largest dimension the solution has not been evaluated yet
nxl, nxu = norm_bounds(current, self.problem)
k = np.argmax(nxu - nxl)
# the delta value to be used to get left and right - this is one sixth of the range
xl, xu = current.get("xl"), current.get("xu")
delta = (xu[k] - xl[k]) / 6
# create the left individual
left_x = np.copy(current.X)
left_x[k] = xl[k] + delta
left = Individual(X=left_x)
# create the right individual
right_x = np.copy(current.X)
right_x[k] = xu[k] - delta
right = Individual(X=right_x)
# update the boundaries for all the points accordingly
for ind in [current, left, right]:
update_bounds(ind, xl, xu, k, delta)
# create the offspring population, evaluate and attach to current population
_infill = Population.create(left, right)
_infill.set("depth", current.get("depth") + 1)
infills = Population.merge(infills, _infill)
return infills
def _backward(self, X, inplace=False, **kwargs):
assert isinstance(X, np.ndarray)
# if it should be 2d we convert it to be a 1-d array
if X.ndim == 2 and len(X) == 1:
vals = X[0]
assert vals.ndim == 1
if inplace:
self.obj.set(self.attr, vals)
return self.obj
else:
return Individual(**{self.attr: vals})
def _initialize_infill(self):
# the boundaries of the problem for initialization
xl, xu = self.problem.bounds()
# take care of the lower bound - make sure it is an individual and make sure it is evaluated
if self.a is None:
assert xl is not None, "Either provide a left bound or set the xl attribute in the problem."
self.a = Individual(X=xl)
if self.a.F is None:
self.evaluator.eval(self.problem, self.a, algorithm=self)
# take care of the upper bound - make sure it is an individual and make sure it is evaluated
if self.b is None:
assert xl is not None, "Either provide a right bound or set the xu attribute in the problem."
self.b = Individual(X=xu)
if self.b.F is None:
self.evaluator.eval(self.problem, self.b, algorithm=self)
assert self.a.X[0] <= self.b.X[
0], "The left bound must be smaller than the left bound!"
def step(self):
alpha, max_alpha = self.alpha, self.problem.xu[0]
if alpha > max_alpha:
alpha = max_alpha
infill = Individual(X=np.array([alpha]))
self.evaluator.eval(self.problem, infill)
self.pop = Population.merge(self.pop, infill)[-10:]
if is_better(self.point, infill, eps=0.0) or alpha == max_alpha:
self.termination.force_termination = True
return
self.point = infill
self.alpha *= 2
def step(problem, x, delta, k):
# copy and add delta to the new point
X = np.copy(x)
# if the problem has bounds normalize the delta
# if problem.has_bounds():
# xl, xu = problem.bounds()
# delta *= (xu[k] - xl[k])
# now add to the current solution
X[k] = X[k] + delta[k]
# repair if out of bounds if necessary
X = set_to_bounds_if_outside_by_problem(problem, X)
return Individual(X=X)
def _setup(self, problem, **kwargs):
xl, xu = problem.bounds()
X = denormalize(0.5 * np.ones(problem.n_var), xl, xu)
x0 = Individual(X=X)
x0.set("xl", xl)
x0.set("xu", xu)
x0.set("depth", 0)
self.x0 = x0
def _advance(self, **kwargs):
# all the elements in the interval
a, b, c = self.pop
# if this is the case then the function is not convex (which means U shaped)
if c.F[0] >= a.F[0] or c.F[0] >= b.F[0]:
# choose the left side if a smaller than b, or the right side otherwise
if a.F[0] <= b.F[0]:
a = c
else:
b = c
c = Individual(X=(b.X - a.X) / 2)
self.evaluator.eval(self.problem, c, algorithm=self)
self.infills = c
else:
d = quadr_interp(a, b, c)
self.evaluator.eval(self.problem, d, algorithm=self)
self.infills = d
# swap c and d -> make sure d is always on the right of c -> a, c, d, b
if c.X[0] > d.X[0]:
c, d = d, c
# if c is better than d, then d becomes the new right bound
if c.F[0] <= d.F[0]:
b = d
# if d is better than c, then c becomes the new left bound and d the new right bound
else:
a, c = c, d
self.pop = Population.create(a, b, c)
def __new__(cls, n_individuals=0):
obj = super(Population, cls).__new__(cls, n_individuals, dtype=cls).view(cls)
for i in range(n_individuals):
obj[i] = Individual()
return obj
def _infill(self):
problem, evaluator = self.problem, self.evaluator
evaluator.skip_already_evaluated = False
sol, direction = self.problem.point, self.problem.direction
# the function value and gradient of the initial solution
sol.set("alpha", 0.0)
sol_F, sol_dF = sol.F[0], sol.get("dF")[0]
def zoom(alpha_low, alpha_high, max_iter=100):
while True:
_alpha = (alpha_high.get("alpha") + alpha_low.get("alpha")) / 2
_point = Individual(X=_alpha)
evaluator.eval(problem, _point, evaluate_values_of=["F", "CV"])
if _point.F[
0] > sol_F + self.c1 * _alpha * sol_dF @ direction or _point.F[
0] > alpha_low.F[0]:
alpha_high = _point
else:
evaluator.eval(problem, _point, evaluate_values_of=["dF"])
point_dF = _point.get("dF")[0]
if np.abs(point_dF
@ direction) <= -self.c2 * sol_dF @ direction:
return _point
if (point_dF @ direction) * (alpha_high.get("alpha") -
alpha_low.get("alpha")) >= 0:
alpha_high = alpha_low
alpha_low = _point
last = sol
alpha = 1.0
current = Individual(X=alpha)
for i in range(1, self.max_iter + 1):
# evaluate the solutions
evaluator.eval(problem, current, evaluate_values_of=["F", "CV"])
# get the values from the solution to be used to evaluate the conditions
F, dF, _F = last.F[0], last.get("dF")[0], current.F[0]
# if the wolfe condition is violate we have found our upper bound
if _F > sol_F + self.c1 * sol_dF @ direction or (i > 1
and F >= _F):
return zoom(last, current)
# for the other condition we need the gradient information
evaluator.eval(problem, current, evaluate_values_of=["dF"])
_dF = current.get("dF")[0]
if np.abs(_dF @ direction) <= -self.c2 * sol_dF @ direction:
return current
if _dF @ direction >= 0:
return zoom(current, last)
alpha = 2 * alpha
last = current
current = Individual(X=alpha)
return current
def wolfe_line_search(problem,
sol,
direction,
c1=1e-4,
c2=0.9,
max_iter=10,
evaluator=None):
# initialize the evaluator to be used (this will make sure evaluations are counted)
evaluator = evaluator if evaluator is not None else Evaluator()
evaluator.skip_already_evaluated = False
# the function value and gradient of the initial solution
sol.set("alpha", 0.0)
sol_F, sol_dF = sol.F[0], sol.get("dF")[0]
def zoom(alpha_low, alpha_high, max_iter=100):
while True:
_alpha = (alpha_high.get("alpha") + alpha_low.get("alpha")) / 2
_point = Individual(X=sol.X + _alpha * direction, alpha=_alpha)
evaluator.eval(problem, _point, evaluate_values_of=["F", "CV"])
if _point.F[
0] > sol_F + c1 * _alpha * sol_dF @ direction or _point.F[
0] > alpha_low.F[0]:
alpha_high = _point
else:
evaluator.eval(problem, _point, evaluate_values_of=["dF"])
point_dF = _point.get("dF")[0]
if np.abs(point_dF @ direction) <= -c2 * sol_dF @ direction:
return _point
if (point_dF @ direction) * (alpha_high.get("alpha") -
alpha_low.get("alpha")) >= 0:
alpha_high = alpha_low
alpha_low = _point
last = sol
alpha = 1.0
current = Individual(X=sol.X + alpha * direction, alpha=alpha)
for i in range(1, max_iter + 1):
# evaluate the solutions
evaluator.eval(problem, current, evaluate_values_of=["F", "CV"])
# get the values from the solution to be used to evaluate the conditions
F, dF, _F = last.F[0], last.get("dF")[0], current.F[0]
# if the wolfe condition is violate we have found our upper bound
if _F > sol_F + c1 * sol_dF @ direction or (i > 1 and F >= _F):
return zoom(last, current)
# for the other condition we need the gradient information
evaluator.eval(problem, current, evaluate_values_of=["dF"])
_dF = current.get("dF")[0]
if np.abs(_dF @ direction) <= -c2 * sol_dF @ direction:
return current
if _dF @ direction >= 0:
return zoom(current, last)
alpha = 2 * alpha
last = current
current = Individual(X=sol.X + alpha * direction, alpha=alpha)
return current
def _local_advance(self, **kwargs):
# number of variables increased by one - matches equations in the paper
xl, xu = self.problem.bounds()
pop, n = self.pop, self.problem.n_var - 1
# calculate the centroid
centroid = pop[:n + 1].get("X").mean(axis=0)
# -------------------------------------------------------------------------------------------
# REFLECT
# -------------------------------------------------------------------------------------------
# check the maximum alpha until the bounds are hit
alphaU = max_alpha(centroid, (centroid - pop[n + 1].X), xl, xu)
# reflect the point, consider factor if bounds are there, make sure in bounds (floating point) evaluate
x_reflect = centroid + min(self.alpha, alphaU) * (centroid - pop[n + 1].X)
x_reflect = set_to_bounds_if_outside_by_problem(self.problem, x_reflect)
reflect = self.evaluator.eval(self.problem, Individual(X=x_reflect), algorithm=self)
# whether a shrink is necessary or not - decided during this step
shrink = False
better_than_current_best = is_better(reflect, pop[0])
better_than_second_worst = is_better(reflect, pop[n])
better_than_worst = is_better(reflect, pop[n + 1])
# if better than the current best - check for expansion
if better_than_current_best:
# -------------------------------------------------------------------------------------------
# EXPAND
# -------------------------------------------------------------------------------------------
# the maximum expansion until the bounds are hit
betaU = max_alpha(centroid, (x_reflect - centroid), xl, xu)
# expand using the factor, consider bounds, make sure in case of floating point issues
x_expand = centroid + min(self.beta, betaU) * (x_reflect - centroid)
x_expand = set_to_bounds_if_outside_by_problem(self.problem, x_expand)
# if the expansion is almost equal to reflection (if boundaries were hit) - no evaluation
if np.allclose(x_expand, x_reflect, atol=1e-16):
expand = reflect
else:
expand = self.evaluator.eval(self.problem, Individual(X=x_expand), algorithm=self)
# if the expansion further improved take it - otherwise use expansion
if is_better(expand, reflect):
pop[n + 1] = expand
else:
pop[n + 1] = reflect
# if the new point is not better than the best, but better than second worst - just keep it
elif not better_than_current_best and better_than_second_worst:
pop[n + 1] = reflect
# if not worse than the worst - outside contraction
elif not better_than_second_worst and better_than_worst:
# -------------------------------------------------------------------------------------------
# Outside Contraction
# -------------------------------------------------------------------------------------------
x_contract_outside = centroid + self.gamma * (x_reflect - centroid)
contract_outside = self.evaluator.eval(self.problem, Individual(X=x_contract_outside), algorithm=self)
if is_better(contract_outside, reflect):
pop[n + 1] = contract_outside
else:
shrink = True
# if the reflection was worse than the worst - inside contraction
else:
# -------------------------------------------------------------------------------------------
# Inside Contraction
# -------------------------------------------------------------------------------------------
x_contract_inside = centroid - self.gamma * (x_reflect - centroid)
contract_inside = self.evaluator.eval(self.problem, Individual(X=x_contract_inside), algorithm=self)
if is_better(contract_inside, pop[n + 1]):
pop[n + 1] = contract_inside
else:
shrink = True
# -------------------------------------------------------------------------------------------
# Shrink (only if necessary)
# -------------------------------------------------------------------------------------------
if shrink:
for i in range(1, len(pop)):
pop[i].X = pop[0].X + self.delta * (pop[i].X - pop[0].X)
pop[1:] = self.evaluator.eval(self.problem, pop[1:], algorithm=self)
self.pop = FitnessSurvival().do(self.problem, pop, n_survive=len(pop))
def test_evaluate_individual():
evaluator = Evaluator()
ind = evaluator.eval(problem, Individual(X=X[0]))
np.testing.assert_allclose(F[0], ind.get("F"))
assert evaluator.n_eval == 1
def _advance(self, **kwargs):
repair, crossover, mutation = self.repair, self.mating.crossover, self.mating.mutation
pc_pop = self.pc_pop.copy(deep=True)
npc_pop = self.npc_pop.copy(deep=True)
##############################################################
# PC evolving
##############################################################
# Normalise both poulations according to the PC individuals
pc_pop, npc_pop = normalize_bothpop(pc_pop, npc_pop)
PCObj = pc_pop.get("F")
NPCObj = npc_pop.get("F")
pc_size = PCObj.shape[0]
npc_size = NPCObj.shape[0]
######################################################
# Calculate the Euclidean distance among individuals
######################################################
d = np.zeros((pc_size, pc_size))
d = cdist(PCObj, PCObj, 'euclidean')
d[d == 0] = np.inf
# Determine the size of the niche
if pc_size == 1:
radius = 0
else:
radius = determine_radius(d, pc_size, self.pc_capacity)
# calculate the radius for individual exploration
r = pc_size / self.pc_capacity * radius
########################################################
# find the promising individuals in PC for exploration
########################################################
# promising_num: record how many promising individuals in PC
promising_num = 0
# count: record how many NPC individuals are located in each PC individual's niche
count = np.array([])
d2 = np.zeros((pc_size, npc_size))
d2 = cdist(PCObj, NPCObj, 'euclidean')
# Count of True elements in each row (each individual in PC) of 2D Numpy Array
count = np.count_nonzero(d2 <= r, axis=1)
# Check if the niche has no NPC individual or has only one NPC individual
# Record the indices of promising individuals.
# Since np.nonzero() returns a tuple of arrays, we change the type of promising_index to a numpy.ndarray.
promising_index = np.nonzero(count <= 1)
promising_index = np.asarray(promising_index).flatten()
# Record total number of promising individuals in PC for exploration
promising_num = len(promising_index)
########################################
# explore these promising individuals
########################################
original_size = pc_size
off = Individual()
if promising_num > 0:
for i in range(promising_num):
if original_size > 1:
parents = Population.new(2)
# The explored individual is considered as one parent
parents[0] = pc_pop[promising_index[i]]
# The remaining parent will be selected randomly from the PC population
rnd = np.random.permutation(pc_size)
for j in rnd:
if j != promising_index[i]:
parents[1] = pc_pop[j]
break
index = np.array([0, 1])
parents_shape = index[None, :]
# do recombination and create an offspring
off = crossover.do(self.problem, parents, parents_shape)[0]
else:
off = pc_pop[0]
# mutation
off = Population.create(off)
off = mutation.do(self.problem, off)
# evaluate the offspring
self.evaluator.eval(self.problem, off)
# update the PC population by the offspring
self.pc_pop = update_PCpop(self.pc_pop, off)
# update the ideal point
self.ideal = np.min(np.vstack([self.ideal,
off.get("F")]),
axis=0)
# update at most one solution in NPC population
self.npc_pop = update_NPCpop(self.npc_pop, off, self.ideal,
self.ref_dirs, self.decomp)
########################################################
# NPC evolution based on MOEA/D
########################################################
# iterate for each member of the population in random order
for i in np.random.permutation(len(self.npc_pop)):
# get the parents using the neighborhood selection
P = self.selection.do(self.npc_pop,
1,
self.mating.crossover.n_parents,
k=[i])
# perform a mating using the default operators (recombination & mutation) - if more than one offspring just pick the first
off = self.mating.do(self.problem, self.npc_pop, 1, parents=P)[0]
off = Population.create(off)
# evaluate the offspring
self.evaluator.eval(self.problem, off, algorithm=self)
# update the PC population by the offspring
self.pc_pop = update_PCpop(self.pc_pop, off)
# update the ideal point
self.ideal = np.min(np.vstack([self.ideal, off.get("F")]), axis=0)
# now actually do the replacement of the individual is better
self.npc_pop = self._replace(i, off)
########################################################
# population maintenance operation in the PC evolution
########################################################
current_pop = Population.merge(self.pc_pop, self.npc_pop)
current_pop = Population.merge(current_pop, self.pop)
# filter duplicate in the population
pc_pop = self.eliminate_duplicates.do(current_pop)
pc_size = len(pc_pop)
if (pc_size > self.pc_capacity):
# get the objective space values and objects
pc_objs = pc_pop.get("F")
fronts, rank = NonDominatedSorting().do(pc_objs, return_rank=True)
front_0_index = fronts[0]
# put the nondominated individuals of the NPC population into the PC population
self.pc_pop = pc_pop[front_0_index]
if len(self.pc_pop) > self.pc_capacity:
self.pc_pop = maintain_PCpop(self.pc_pop, self.pc_capacity)
self.pop = self.pc_pop.copy(deep=True)
def quadr_interp(a, b, c):
return Individual(X=quadr_interp_equ(a.X, a.F, b.X, b.F, c.X, c.F))
def _advance(self, **kwargs):
problem, evaluator = self.problem, self.evaluator
# add the box constraints defined in the problem
bounds = None
if self.use_bounds:
xl, xu = self.problem.bounds()
if self.with_bounds:
bounds = np.column_stack([xl, xu])
else:
if xl is not None or xu is not None:
raise Exception(
f"Error: Boundary constraints can not be handled by {self.method}"
)
# define the actual constraints if supported by the algorithm
constr = []
if self.use_constr:
constr = [LinearConstraint(np.eye(self.problem.n_var), xl, xu)]
if problem.has_constraints():
if self.with_constr:
def fun_constr(x):
g, cv = problem.evaluate(x,
return_values_of=["G", "CV"])
return cv[0]
non_lin_constr = NonlinearConstraint(
fun_constr, -float("inf"), 0)
constr.append(non_lin_constr)
else:
raise Exception(
f"Error: Constraint handling is not supported by {self.method}"
)
# the objective function to be optimized and add gradients if available
if self.estm_gradients:
jac = None
def fun_obj(x):
f = problem.evaluate(x, return_values_of=["F"])[0]
evaluator.n_eval += 1
return f
else:
jac = True
def fun_obj(x):
f, df = problem.evaluate(x, return_values_of=["F", "dF"])
if df is None:
raise Exception(
"If the gradient shall not be estimate, please set out['dF'] in _evaluate. "
)
evaluator.n_eval += 1
return f[0], df[0]
# the arguments to be used
kwargs = dict(args=(),
method=self.method,
bounds=bounds,
constraints=constr,
jac=jac,
options=self.options)
# the starting solution found by sampling beforehand
x0 = self.opt[0].X
# actually run the optimization
if not self.show_warnings:
warnings.simplefilter("ignore")
res = scipy_minimize(fun_obj, x0, **kwargs)
opt = Population.create(Individual(X=res.x))
self.evaluator.eval(self.problem, opt, algorithm=self)
self.pop, self.off = opt, opt
self.termination.force_termination = True
if hasattr("res", "nit"):
self.n_gen = res.nit + 1
