662 lines
21 KiB
Python
662 lines
21 KiB
Python
import numpy as np
|
|
import copy
|
|
|
|
|
|
class Complex:
|
|
def __init__(self, dim, func, func_args=(), symmetry=False, bounds=None,
|
|
g_cons=None, g_args=()):
|
|
self.dim = dim
|
|
self.bounds = bounds
|
|
self.symmetry = symmetry # TODO: Define the functions to be used
|
|
# here in init to avoid if checks
|
|
self.gen = 0
|
|
self.perm_cycle = 0
|
|
|
|
# Every cell is stored in a list of its generation,
|
|
# e.g., the initial cell is stored in self.H[0]
|
|
# 1st get new cells are stored in self.H[1] etc.
|
|
# When a cell is subgenerated it is removed from this list
|
|
|
|
self.H = [] # Storage structure of cells
|
|
# Cache of all vertices
|
|
self.V = VertexCache(func, func_args, bounds, g_cons, g_args)
|
|
|
|
# Generate n-cube here:
|
|
self.n_cube(dim, symmetry=symmetry)
|
|
|
|
# TODO: Assign functions to a the complex instead
|
|
if symmetry:
|
|
self.generation_cycle = 1
|
|
# self.centroid = self.C0()[-1].x
|
|
# self.C0.centroid = self.centroid
|
|
else:
|
|
self.add_centroid()
|
|
|
|
self.H.append([])
|
|
self.H[0].append(self.C0)
|
|
self.hgr = self.C0.homology_group_rank()
|
|
self.hgrd = 0 # Complex group rank differential
|
|
# self.hgr = self.C0.hg_n
|
|
|
|
# Build initial graph
|
|
self.graph_map()
|
|
|
|
self.performance = []
|
|
self.performance.append(0)
|
|
self.performance.append(0)
|
|
|
|
def __call__(self):
|
|
return self.H
|
|
|
|
def n_cube(self, dim, symmetry=False, printout=False):
|
|
"""
|
|
Generate the simplicial triangulation of the N-D hypercube
|
|
containing 2**n vertices
|
|
"""
|
|
origin = list(np.zeros(dim, dtype=int))
|
|
self.origin = origin
|
|
supremum = list(np.ones(dim, dtype=int))
|
|
self.supremum = supremum
|
|
|
|
# tuple versions for indexing
|
|
origintuple = tuple(origin)
|
|
supremumtuple = tuple(supremum)
|
|
|
|
x_parents = [origintuple]
|
|
|
|
if symmetry:
|
|
self.C0 = Simplex(0, 0, 0, self.dim) # Initial cell object
|
|
self.C0.add_vertex(self.V[origintuple])
|
|
|
|
i_s = 0
|
|
self.perm_symmetry(i_s, x_parents, origin)
|
|
self.C0.add_vertex(self.V[supremumtuple])
|
|
else:
|
|
self.C0 = Cell(0, 0, origin, supremum) # Initial cell object
|
|
self.C0.add_vertex(self.V[origintuple])
|
|
self.C0.add_vertex(self.V[supremumtuple])
|
|
|
|
i_parents = []
|
|
self.perm(i_parents, x_parents, origin)
|
|
|
|
if printout:
|
|
print("Initial hyper cube:")
|
|
for v in self.C0():
|
|
v.print_out()
|
|
|
|
def perm(self, i_parents, x_parents, xi):
|
|
# TODO: Cut out of for if outside linear constraint cutting planes
|
|
xi_t = tuple(xi)
|
|
|
|
# Construct required iterator
|
|
iter_range = [x for x in range(self.dim) if x not in i_parents]
|
|
|
|
for i in iter_range:
|
|
i2_parents = copy.copy(i_parents)
|
|
i2_parents.append(i)
|
|
xi2 = copy.copy(xi)
|
|
xi2[i] = 1
|
|
# Make new vertex list a hashable tuple
|
|
xi2_t = tuple(xi2)
|
|
# Append to cell
|
|
self.C0.add_vertex(self.V[xi2_t])
|
|
# Connect neighbors and vice versa
|
|
# Parent point
|
|
self.V[xi2_t].connect(self.V[xi_t])
|
|
|
|
# Connect all family of simplices in parent containers
|
|
for x_ip in x_parents:
|
|
self.V[xi2_t].connect(self.V[x_ip])
|
|
|
|
x_parents2 = copy.copy(x_parents)
|
|
x_parents2.append(xi_t)
|
|
|
|
# Permutate
|
|
self.perm(i2_parents, x_parents2, xi2)
|
|
|
|
def perm_symmetry(self, i_s, x_parents, xi):
|
|
# TODO: Cut out of for if outside linear constraint cutting planes
|
|
xi_t = tuple(xi)
|
|
xi2 = copy.copy(xi)
|
|
xi2[i_s] = 1
|
|
# Make new vertex list a hashable tuple
|
|
xi2_t = tuple(xi2)
|
|
# Append to cell
|
|
self.C0.add_vertex(self.V[xi2_t])
|
|
# Connect neighbors and vice versa
|
|
# Parent point
|
|
self.V[xi2_t].connect(self.V[xi_t])
|
|
|
|
# Connect all family of simplices in parent containers
|
|
for x_ip in x_parents:
|
|
self.V[xi2_t].connect(self.V[x_ip])
|
|
|
|
x_parents2 = copy.copy(x_parents)
|
|
x_parents2.append(xi_t)
|
|
|
|
i_s += 1
|
|
if i_s == self.dim:
|
|
return
|
|
# Permutate
|
|
self.perm_symmetry(i_s, x_parents2, xi2)
|
|
|
|
def add_centroid(self):
|
|
"""Split the central edge between the origin and supremum of
|
|
a cell and add the new vertex to the complex"""
|
|
self.centroid = list(
|
|
(np.array(self.origin) + np.array(self.supremum)) / 2.0)
|
|
self.C0.add_vertex(self.V[tuple(self.centroid)])
|
|
self.C0.centroid = self.centroid
|
|
|
|
# Disconnect origin and supremum
|
|
self.V[tuple(self.origin)].disconnect(self.V[tuple(self.supremum)])
|
|
|
|
# Connect centroid to all other vertices
|
|
for v in self.C0():
|
|
self.V[tuple(self.centroid)].connect(self.V[tuple(v.x)])
|
|
|
|
self.centroid_added = True
|
|
return
|
|
|
|
# Construct incidence array:
|
|
def incidence(self):
|
|
if self.centroid_added:
|
|
self.structure = np.zeros([2 ** self.dim + 1, 2 ** self.dim + 1],
|
|
dtype=int)
|
|
else:
|
|
self.structure = np.zeros([2 ** self.dim, 2 ** self.dim],
|
|
dtype=int)
|
|
|
|
for v in self.HC.C0():
|
|
for v2 in v.nn:
|
|
self.structure[v.index, v2.index] = 1
|
|
|
|
return
|
|
|
|
# A more sparse incidence generator:
|
|
def graph_map(self):
|
|
""" Make a list of size 2**n + 1 where an entry is a vertex
|
|
incidence, each list element contains a list of indexes
|
|
corresponding to that entries neighbors"""
|
|
|
|
self.graph = [[v2.index for v2 in v.nn] for v in self.C0()]
|
|
|
|
# Graph structure method:
|
|
# 0. Capture the indices of the initial cell.
|
|
# 1. Generate new origin and supremum scalars based on current generation
|
|
# 2. Generate a new set of vertices corresponding to a new
|
|
# "origin" and "supremum"
|
|
# 3. Connected based on the indices of the previous graph structure
|
|
# 4. Disconnect the edges in the original cell
|
|
|
|
def sub_generate_cell(self, C_i, gen):
|
|
"""Subgenerate a cell `C_i` of generation `gen` and
|
|
homology group rank `hgr`."""
|
|
origin_new = tuple(C_i.centroid)
|
|
centroid_index = len(C_i()) - 1
|
|
|
|
# If not gen append
|
|
try:
|
|
self.H[gen]
|
|
except IndexError:
|
|
self.H.append([])
|
|
|
|
# Generate subcubes using every extreme vertex in C_i as a supremum
|
|
# and the centroid of C_i as the origin
|
|
H_new = [] # list storing all the new cubes split from C_i
|
|
for i, v in enumerate(C_i()[:-1]):
|
|
supremum = tuple(v.x)
|
|
H_new.append(
|
|
self.construct_hypercube(origin_new, supremum, gen, C_i.hg_n))
|
|
|
|
for i, connections in enumerate(self.graph):
|
|
# Present vertex V_new[i]; connect to all connections:
|
|
if i == centroid_index: # Break out of centroid
|
|
break
|
|
|
|
for j in connections:
|
|
C_i()[i].disconnect(C_i()[j])
|
|
|
|
# Destroy the old cell
|
|
if C_i is not self.C0: # Garbage collector does this anyway; not needed
|
|
del C_i
|
|
|
|
# TODO: Recalculate all the homology group ranks of each cell
|
|
return H_new
|
|
|
|
def split_generation(self):
|
|
"""
|
|
Run sub_generate_cell for every cell in the current complex self.gen
|
|
"""
|
|
no_splits = False # USED IN SHGO
|
|
try:
|
|
for c in self.H[self.gen]:
|
|
if self.symmetry:
|
|
# self.sub_generate_cell_symmetry(c, self.gen + 1)
|
|
self.split_simplex_symmetry(c, self.gen + 1)
|
|
else:
|
|
self.sub_generate_cell(c, self.gen + 1)
|
|
except IndexError:
|
|
no_splits = True # USED IN SHGO
|
|
|
|
self.gen += 1
|
|
return no_splits # USED IN SHGO
|
|
|
|
def construct_hypercube(self, origin, supremum, gen, hgr,
|
|
printout=False):
|
|
"""
|
|
Build a hypercube with triangulations symmetric to C0.
|
|
|
|
Parameters
|
|
----------
|
|
origin : vec
|
|
supremum : vec (tuple)
|
|
gen : generation
|
|
hgr : parent homology group rank
|
|
"""
|
|
# Initiate new cell
|
|
v_o = np.array(origin)
|
|
v_s = np.array(supremum)
|
|
|
|
C_new = Cell(gen, hgr, origin, supremum)
|
|
C_new.centroid = tuple((v_o + v_s) * .5)
|
|
|
|
# Build new indexed vertex list
|
|
V_new = []
|
|
|
|
for i, v in enumerate(self.C0()[:-1]):
|
|
v_x = np.array(v.x)
|
|
sub_cell_t1 = v_o - v_o * v_x
|
|
sub_cell_t2 = v_s * v_x
|
|
|
|
vec = sub_cell_t1 + sub_cell_t2
|
|
|
|
vec = tuple(vec)
|
|
C_new.add_vertex(self.V[vec])
|
|
V_new.append(vec)
|
|
|
|
# Add new centroid
|
|
C_new.add_vertex(self.V[C_new.centroid])
|
|
V_new.append(C_new.centroid)
|
|
|
|
# Connect new vertices #TODO: Thread into other loop; no need for V_new
|
|
for i, connections in enumerate(self.graph):
|
|
# Present vertex V_new[i]; connect to all connections:
|
|
for j in connections:
|
|
self.V[V_new[i]].connect(self.V[V_new[j]])
|
|
|
|
if printout:
|
|
print("A sub hyper cube with:")
|
|
print("origin: {}".format(origin))
|
|
print("supremum: {}".format(supremum))
|
|
for v in C_new():
|
|
v.print_out()
|
|
|
|
# Append the new cell to the to complex
|
|
self.H[gen].append(C_new)
|
|
|
|
return C_new
|
|
|
|
def split_simplex_symmetry(self, S, gen):
|
|
"""
|
|
Split a hypersimplex S into two sub simplices by building a hyperplane
|
|
which connects to a new vertex on an edge (the longest edge in
|
|
dim = {2, 3}) and every other vertex in the simplex that is not
|
|
connected to the edge being split.
|
|
|
|
This function utilizes the knowledge that the problem is specified
|
|
with symmetric constraints
|
|
|
|
The longest edge is tracked by an ordering of the
|
|
vertices in every simplices, the edge between first and second
|
|
vertex is the longest edge to be split in the next iteration.
|
|
"""
|
|
# If not gen append
|
|
try:
|
|
self.H[gen]
|
|
except IndexError:
|
|
self.H.append([])
|
|
|
|
# Find new vertex.
|
|
# V_new_x = tuple((np.array(C()[0].x) + np.array(C()[1].x)) / 2.0)
|
|
s = S()
|
|
firstx = s[0].x
|
|
lastx = s[-1].x
|
|
V_new = self.V[tuple((np.array(firstx) + np.array(lastx)) / 2.0)]
|
|
|
|
# Disconnect old longest edge
|
|
self.V[firstx].disconnect(self.V[lastx])
|
|
|
|
# Connect new vertices to all other vertices
|
|
for v in s[:]:
|
|
v.connect(self.V[V_new.x])
|
|
|
|
# New "lower" simplex
|
|
S_new_l = Simplex(gen, S.hg_n, self.generation_cycle,
|
|
self.dim)
|
|
S_new_l.add_vertex(s[0])
|
|
S_new_l.add_vertex(V_new) # Add new vertex
|
|
for v in s[1:-1]: # Add all other vertices
|
|
S_new_l.add_vertex(v)
|
|
|
|
# New "upper" simplex
|
|
S_new_u = Simplex(gen, S.hg_n, S.generation_cycle, self.dim)
|
|
|
|
# First vertex on new long edge
|
|
S_new_u.add_vertex(s[S_new_u.generation_cycle + 1])
|
|
|
|
for v in s[1:-1]: # Remaining vertices
|
|
S_new_u.add_vertex(v)
|
|
|
|
for k, v in enumerate(s[1:-1]): # iterate through inner vertices
|
|
if k == S.generation_cycle:
|
|
S_new_u.add_vertex(V_new)
|
|
else:
|
|
S_new_u.add_vertex(v)
|
|
|
|
S_new_u.add_vertex(s[-1]) # Second vertex on new long edge
|
|
|
|
self.H[gen].append(S_new_l)
|
|
self.H[gen].append(S_new_u)
|
|
|
|
return
|
|
|
|
# Plots
|
|
def plot_complex(self):
|
|
"""
|
|
Here, C is the LIST of simplexes S in the
|
|
2- or 3-D complex
|
|
|
|
To plot a single simplex S in a set C, use e.g., [C[0]]
|
|
"""
|
|
from matplotlib import pyplot # type: ignore[import]
|
|
if self.dim == 2:
|
|
pyplot.figure()
|
|
for C in self.H:
|
|
for c in C:
|
|
for v in c():
|
|
if self.bounds is None:
|
|
x_a = np.array(v.x, dtype=float)
|
|
else:
|
|
x_a = np.array(v.x, dtype=float)
|
|
for i in range(len(self.bounds)):
|
|
x_a[i] = (x_a[i] * (self.bounds[i][1]
|
|
- self.bounds[i][0])
|
|
+ self.bounds[i][0])
|
|
|
|
# logging.info('v.x_a = {}'.format(x_a))
|
|
|
|
pyplot.plot([x_a[0]], [x_a[1]], 'o')
|
|
|
|
xlines = []
|
|
ylines = []
|
|
for vn in v.nn:
|
|
if self.bounds is None:
|
|
xn_a = np.array(vn.x, dtype=float)
|
|
else:
|
|
xn_a = np.array(vn.x, dtype=float)
|
|
for i in range(len(self.bounds)):
|
|
xn_a[i] = (xn_a[i] * (self.bounds[i][1]
|
|
- self.bounds[i][0])
|
|
+ self.bounds[i][0])
|
|
|
|
# logging.info('vn.x = {}'.format(vn.x))
|
|
|
|
xlines.append(xn_a[0])
|
|
ylines.append(xn_a[1])
|
|
xlines.append(x_a[0])
|
|
ylines.append(x_a[1])
|
|
|
|
pyplot.plot(xlines, ylines)
|
|
|
|
if self.bounds is None:
|
|
pyplot.ylim([-1e-2, 1 + 1e-2])
|
|
pyplot.xlim([-1e-2, 1 + 1e-2])
|
|
else:
|
|
pyplot.ylim(
|
|
[self.bounds[1][0] - 1e-2, self.bounds[1][1] + 1e-2])
|
|
pyplot.xlim(
|
|
[self.bounds[0][0] - 1e-2, self.bounds[0][1] + 1e-2])
|
|
|
|
pyplot.show()
|
|
|
|
elif self.dim == 3:
|
|
fig = pyplot.figure()
|
|
ax = fig.add_subplot(111, projection='3d')
|
|
|
|
for C in self.H:
|
|
for c in C:
|
|
for v in c():
|
|
x = []
|
|
y = []
|
|
z = []
|
|
# logging.info('v.x = {}'.format(v.x))
|
|
x.append(v.x[0])
|
|
y.append(v.x[1])
|
|
z.append(v.x[2])
|
|
for vn in v.nn:
|
|
x.append(vn.x[0])
|
|
y.append(vn.x[1])
|
|
z.append(vn.x[2])
|
|
x.append(v.x[0])
|
|
y.append(v.x[1])
|
|
z.append(v.x[2])
|
|
# logging.info('vn.x = {}'.format(vn.x))
|
|
|
|
ax.plot(x, y, z, label='simplex')
|
|
|
|
pyplot.show()
|
|
else:
|
|
print("dimension higher than 3 or wrong complex format")
|
|
return
|
|
|
|
|
|
class VertexGroup(object):
|
|
def __init__(self, p_gen, p_hgr):
|
|
self.p_gen = p_gen # parent generation
|
|
self.p_hgr = p_hgr # parent homology group rank
|
|
self.hg_n = None
|
|
self.hg_d = None
|
|
|
|
# Maybe add parent homology group rank total history
|
|
# This is the sum off all previously split cells
|
|
# cumulatively throughout its entire history
|
|
self.C = []
|
|
|
|
def __call__(self):
|
|
return self.C
|
|
|
|
def add_vertex(self, V):
|
|
if V not in self.C:
|
|
self.C.append(V)
|
|
|
|
def homology_group_rank(self):
|
|
"""
|
|
Returns the homology group order of the current cell
|
|
"""
|
|
if self.hg_n is None:
|
|
self.hg_n = sum(1 for v in self.C if v.minimiser())
|
|
|
|
return self.hg_n
|
|
|
|
def homology_group_differential(self):
|
|
"""
|
|
Returns the difference between the current homology group of the
|
|
cell and its parent group
|
|
"""
|
|
if self.hg_d is None:
|
|
self.hgd = self.hg_n - self.p_hgr
|
|
|
|
return self.hgd
|
|
|
|
def polytopial_sperner_lemma(self):
|
|
"""
|
|
Returns the number of stationary points theoretically contained in the
|
|
cell based information currently known about the cell
|
|
"""
|
|
pass
|
|
|
|
def print_out(self):
|
|
"""
|
|
Print the current cell to console
|
|
"""
|
|
for v in self():
|
|
v.print_out()
|
|
|
|
|
|
class Cell(VertexGroup):
|
|
"""
|
|
Contains a cell that is symmetric to the initial hypercube triangulation
|
|
"""
|
|
|
|
def __init__(self, p_gen, p_hgr, origin, supremum):
|
|
super(Cell, self).__init__(p_gen, p_hgr)
|
|
|
|
self.origin = origin
|
|
self.supremum = supremum
|
|
self.centroid = None # (Not always used)
|
|
# TODO: self.bounds
|
|
|
|
|
|
class Simplex(VertexGroup):
|
|
"""
|
|
Contains a simplex that is symmetric to the initial symmetry constrained
|
|
hypersimplex triangulation
|
|
"""
|
|
|
|
def __init__(self, p_gen, p_hgr, generation_cycle, dim):
|
|
super(Simplex, self).__init__(p_gen, p_hgr)
|
|
|
|
self.generation_cycle = (generation_cycle + 1) % (dim - 1)
|
|
|
|
|
|
class Vertex:
|
|
def __init__(self, x, bounds=None, func=None, func_args=(), g_cons=None,
|
|
g_cons_args=(), nn=None, index=None):
|
|
self.x = x
|
|
self.order = sum(x)
|
|
x_a = np.array(x, dtype=float)
|
|
if bounds is not None:
|
|
for i, (lb, ub) in enumerate(bounds):
|
|
x_a[i] = x_a[i] * (ub - lb) + lb
|
|
|
|
# TODO: Make saving the array structure optional
|
|
self.x_a = x_a
|
|
|
|
# Note Vertex is only initiated once for all x so only
|
|
# evaluated once
|
|
if func is not None:
|
|
self.feasible = True
|
|
if g_cons is not None:
|
|
for g, args in zip(g_cons, g_cons_args):
|
|
if g(self.x_a, *args) < 0.0:
|
|
self.f = np.inf
|
|
self.feasible = False
|
|
break
|
|
if self.feasible:
|
|
self.f = func(x_a, *func_args)
|
|
|
|
if nn is not None:
|
|
self.nn = nn
|
|
else:
|
|
self.nn = set()
|
|
|
|
self.fval = None
|
|
self.check_min = True
|
|
|
|
# Index:
|
|
if index is not None:
|
|
self.index = index
|
|
|
|
def __hash__(self):
|
|
return hash(self.x)
|
|
|
|
def connect(self, v):
|
|
if v is not self and v not in self.nn:
|
|
self.nn.add(v)
|
|
v.nn.add(self)
|
|
|
|
if self.minimiser():
|
|
v._min = False
|
|
v.check_min = False
|
|
|
|
# TEMPORARY
|
|
self.check_min = True
|
|
v.check_min = True
|
|
|
|
def disconnect(self, v):
|
|
if v in self.nn:
|
|
self.nn.remove(v)
|
|
v.nn.remove(self)
|
|
self.check_min = True
|
|
v.check_min = True
|
|
|
|
def minimiser(self):
|
|
"""Check whether this vertex is strictly less than all its neighbors"""
|
|
if self.check_min:
|
|
self._min = all(self.f < v.f for v in self.nn)
|
|
self.check_min = False
|
|
|
|
return self._min
|
|
|
|
def print_out(self):
|
|
print("Vertex: {}".format(self.x))
|
|
constr = 'Connections: '
|
|
for vc in self.nn:
|
|
constr += '{} '.format(vc.x)
|
|
|
|
print(constr)
|
|
print('Order = {}'.format(self.order))
|
|
|
|
|
|
class VertexCache:
|
|
def __init__(self, func, func_args=(), bounds=None, g_cons=None,
|
|
g_cons_args=(), indexed=True):
|
|
|
|
self.cache = {}
|
|
self.func = func
|
|
self.g_cons = g_cons
|
|
self.g_cons_args = g_cons_args
|
|
self.func_args = func_args
|
|
self.bounds = bounds
|
|
self.nfev = 0
|
|
self.size = 0
|
|
|
|
if indexed:
|
|
self.index = -1
|
|
|
|
def __getitem__(self, x, indexed=True):
|
|
try:
|
|
return self.cache[x]
|
|
except KeyError:
|
|
if indexed:
|
|
self.index += 1
|
|
xval = Vertex(x, bounds=self.bounds,
|
|
func=self.func, func_args=self.func_args,
|
|
g_cons=self.g_cons,
|
|
g_cons_args=self.g_cons_args,
|
|
index=self.index)
|
|
else:
|
|
xval = Vertex(x, bounds=self.bounds,
|
|
func=self.func, func_args=self.func_args,
|
|
g_cons=self.g_cons,
|
|
g_cons_args=self.g_cons_args)
|
|
|
|
# logging.info("New generated vertex at x = {}".format(x))
|
|
# NOTE: Surprisingly high performance increase if logging is commented out
|
|
self.cache[x] = xval
|
|
|
|
# TODO: Check
|
|
if self.func is not None:
|
|
if self.g_cons is not None:
|
|
if xval.feasible:
|
|
self.nfev += 1
|
|
self.size += 1
|
|
else:
|
|
self.size += 1
|
|
else:
|
|
self.nfev += 1
|
|
self.size += 1
|
|
|
|
return self.cache[x]
|