662 lines
21 KiB
Python
662 lines
21 KiB
Python
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import numpy as np
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import copy
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class Complex:
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def __init__(self, dim, func, func_args=(), symmetry=False, bounds=None,
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g_cons=None, g_args=()):
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self.dim = dim
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self.bounds = bounds
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self.symmetry = symmetry # TODO: Define the functions to be used
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# here in init to avoid if checks
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self.gen = 0
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self.perm_cycle = 0
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# Every cell is stored in a list of its generation,
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# e.g., the initial cell is stored in self.H[0]
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# 1st get new cells are stored in self.H[1] etc.
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# When a cell is subgenerated it is removed from this list
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self.H = [] # Storage structure of cells
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# Cache of all vertices
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self.V = VertexCache(func, func_args, bounds, g_cons, g_args)
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# Generate n-cube here:
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self.n_cube(dim, symmetry=symmetry)
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# TODO: Assign functions to a the complex instead
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if symmetry:
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self.generation_cycle = 1
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# self.centroid = self.C0()[-1].x
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# self.C0.centroid = self.centroid
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else:
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self.add_centroid()
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self.H.append([])
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self.H[0].append(self.C0)
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self.hgr = self.C0.homology_group_rank()
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self.hgrd = 0 # Complex group rank differential
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# self.hgr = self.C0.hg_n
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# Build initial graph
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self.graph_map()
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self.performance = []
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self.performance.append(0)
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self.performance.append(0)
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def __call__(self):
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return self.H
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def n_cube(self, dim, symmetry=False, printout=False):
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"""
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Generate the simplicial triangulation of the N-D hypercube
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containing 2**n vertices
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"""
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origin = list(np.zeros(dim, dtype=int))
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self.origin = origin
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supremum = list(np.ones(dim, dtype=int))
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self.supremum = supremum
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# tuple versions for indexing
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origintuple = tuple(origin)
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supremumtuple = tuple(supremum)
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x_parents = [origintuple]
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if symmetry:
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self.C0 = Simplex(0, 0, 0, self.dim) # Initial cell object
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self.C0.add_vertex(self.V[origintuple])
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i_s = 0
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self.perm_symmetry(i_s, x_parents, origin)
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self.C0.add_vertex(self.V[supremumtuple])
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else:
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self.C0 = Cell(0, 0, origin, supremum) # Initial cell object
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self.C0.add_vertex(self.V[origintuple])
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self.C0.add_vertex(self.V[supremumtuple])
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i_parents = []
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self.perm(i_parents, x_parents, origin)
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if printout:
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print("Initial hyper cube:")
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for v in self.C0():
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v.print_out()
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def perm(self, i_parents, x_parents, xi):
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# TODO: Cut out of for if outside linear constraint cutting planes
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xi_t = tuple(xi)
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# Construct required iterator
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iter_range = [x for x in range(self.dim) if x not in i_parents]
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for i in iter_range:
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i2_parents = copy.copy(i_parents)
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i2_parents.append(i)
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xi2 = copy.copy(xi)
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xi2[i] = 1
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# Make new vertex list a hashable tuple
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xi2_t = tuple(xi2)
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# Append to cell
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self.C0.add_vertex(self.V[xi2_t])
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# Connect neighbors and vice versa
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# Parent point
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self.V[xi2_t].connect(self.V[xi_t])
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# Connect all family of simplices in parent containers
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for x_ip in x_parents:
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self.V[xi2_t].connect(self.V[x_ip])
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x_parents2 = copy.copy(x_parents)
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x_parents2.append(xi_t)
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# Permutate
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self.perm(i2_parents, x_parents2, xi2)
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def perm_symmetry(self, i_s, x_parents, xi):
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# TODO: Cut out of for if outside linear constraint cutting planes
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xi_t = tuple(xi)
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xi2 = copy.copy(xi)
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xi2[i_s] = 1
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# Make new vertex list a hashable tuple
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xi2_t = tuple(xi2)
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# Append to cell
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self.C0.add_vertex(self.V[xi2_t])
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# Connect neighbors and vice versa
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# Parent point
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self.V[xi2_t].connect(self.V[xi_t])
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# Connect all family of simplices in parent containers
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for x_ip in x_parents:
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self.V[xi2_t].connect(self.V[x_ip])
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x_parents2 = copy.copy(x_parents)
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x_parents2.append(xi_t)
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i_s += 1
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if i_s == self.dim:
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return
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# Permutate
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self.perm_symmetry(i_s, x_parents2, xi2)
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def add_centroid(self):
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"""Split the central edge between the origin and supremum of
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a cell and add the new vertex to the complex"""
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self.centroid = list(
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(np.array(self.origin) + np.array(self.supremum)) / 2.0)
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self.C0.add_vertex(self.V[tuple(self.centroid)])
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self.C0.centroid = self.centroid
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# Disconnect origin and supremum
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self.V[tuple(self.origin)].disconnect(self.V[tuple(self.supremum)])
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# Connect centroid to all other vertices
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for v in self.C0():
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self.V[tuple(self.centroid)].connect(self.V[tuple(v.x)])
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self.centroid_added = True
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return
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# Construct incidence array:
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def incidence(self):
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if self.centroid_added:
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self.structure = np.zeros([2 ** self.dim + 1, 2 ** self.dim + 1],
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dtype=int)
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else:
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self.structure = np.zeros([2 ** self.dim, 2 ** self.dim],
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dtype=int)
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for v in self.HC.C0():
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for v2 in v.nn:
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self.structure[v.index, v2.index] = 1
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return
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# A more sparse incidence generator:
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def graph_map(self):
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""" Make a list of size 2**n + 1 where an entry is a vertex
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incidence, each list element contains a list of indexes
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corresponding to that entries neighbors"""
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self.graph = [[v2.index for v2 in v.nn] for v in self.C0()]
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# Graph structure method:
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# 0. Capture the indices of the initial cell.
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# 1. Generate new origin and supremum scalars based on current generation
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# 2. Generate a new set of vertices corresponding to a new
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# "origin" and "supremum"
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# 3. Connected based on the indices of the previous graph structure
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# 4. Disconnect the edges in the original cell
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def sub_generate_cell(self, C_i, gen):
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"""Subgenerate a cell `C_i` of generation `gen` and
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homology group rank `hgr`."""
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origin_new = tuple(C_i.centroid)
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centroid_index = len(C_i()) - 1
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# If not gen append
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try:
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self.H[gen]
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except IndexError:
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self.H.append([])
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# Generate subcubes using every extreme vertex in C_i as a supremum
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# and the centroid of C_i as the origin
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H_new = [] # list storing all the new cubes split from C_i
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for i, v in enumerate(C_i()[:-1]):
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supremum = tuple(v.x)
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H_new.append(
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self.construct_hypercube(origin_new, supremum, gen, C_i.hg_n))
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for i, connections in enumerate(self.graph):
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# Present vertex V_new[i]; connect to all connections:
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if i == centroid_index: # Break out of centroid
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break
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for j in connections:
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C_i()[i].disconnect(C_i()[j])
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# Destroy the old cell
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if C_i is not self.C0: # Garbage collector does this anyway; not needed
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del C_i
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# TODO: Recalculate all the homology group ranks of each cell
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return H_new
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def split_generation(self):
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"""
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Run sub_generate_cell for every cell in the current complex self.gen
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"""
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no_splits = False # USED IN SHGO
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try:
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for c in self.H[self.gen]:
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if self.symmetry:
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# self.sub_generate_cell_symmetry(c, self.gen + 1)
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self.split_simplex_symmetry(c, self.gen + 1)
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else:
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self.sub_generate_cell(c, self.gen + 1)
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except IndexError:
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no_splits = True # USED IN SHGO
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self.gen += 1
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return no_splits # USED IN SHGO
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def construct_hypercube(self, origin, supremum, gen, hgr,
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printout=False):
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"""
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Build a hypercube with triangulations symmetric to C0.
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Parameters
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----------
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origin : vec
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supremum : vec (tuple)
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gen : generation
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hgr : parent homology group rank
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"""
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# Initiate new cell
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v_o = np.array(origin)
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v_s = np.array(supremum)
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C_new = Cell(gen, hgr, origin, supremum)
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C_new.centroid = tuple((v_o + v_s) * .5)
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# Build new indexed vertex list
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V_new = []
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for i, v in enumerate(self.C0()[:-1]):
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v_x = np.array(v.x)
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sub_cell_t1 = v_o - v_o * v_x
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sub_cell_t2 = v_s * v_x
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vec = sub_cell_t1 + sub_cell_t2
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vec = tuple(vec)
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C_new.add_vertex(self.V[vec])
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V_new.append(vec)
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# Add new centroid
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C_new.add_vertex(self.V[C_new.centroid])
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V_new.append(C_new.centroid)
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# Connect new vertices #TODO: Thread into other loop; no need for V_new
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for i, connections in enumerate(self.graph):
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# Present vertex V_new[i]; connect to all connections:
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for j in connections:
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self.V[V_new[i]].connect(self.V[V_new[j]])
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if printout:
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print("A sub hyper cube with:")
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print("origin: {}".format(origin))
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print("supremum: {}".format(supremum))
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for v in C_new():
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v.print_out()
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# Append the new cell to the to complex
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self.H[gen].append(C_new)
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return C_new
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def split_simplex_symmetry(self, S, gen):
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"""
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Split a hypersimplex S into two sub simplices by building a hyperplane
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which connects to a new vertex on an edge (the longest edge in
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dim = {2, 3}) and every other vertex in the simplex that is not
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connected to the edge being split.
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This function utilizes the knowledge that the problem is specified
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with symmetric constraints
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The longest edge is tracked by an ordering of the
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vertices in every simplices, the edge between first and second
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vertex is the longest edge to be split in the next iteration.
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"""
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# If not gen append
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try:
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self.H[gen]
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except IndexError:
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self.H.append([])
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# Find new vertex.
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# V_new_x = tuple((np.array(C()[0].x) + np.array(C()[1].x)) / 2.0)
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s = S()
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firstx = s[0].x
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lastx = s[-1].x
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V_new = self.V[tuple((np.array(firstx) + np.array(lastx)) / 2.0)]
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# Disconnect old longest edge
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self.V[firstx].disconnect(self.V[lastx])
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# Connect new vertices to all other vertices
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for v in s[:]:
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v.connect(self.V[V_new.x])
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# New "lower" simplex
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S_new_l = Simplex(gen, S.hg_n, self.generation_cycle,
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self.dim)
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S_new_l.add_vertex(s[0])
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S_new_l.add_vertex(V_new) # Add new vertex
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for v in s[1:-1]: # Add all other vertices
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S_new_l.add_vertex(v)
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# New "upper" simplex
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S_new_u = Simplex(gen, S.hg_n, S.generation_cycle, self.dim)
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# First vertex on new long edge
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S_new_u.add_vertex(s[S_new_u.generation_cycle + 1])
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for v in s[1:-1]: # Remaining vertices
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S_new_u.add_vertex(v)
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for k, v in enumerate(s[1:-1]): # iterate through inner vertices
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if k == S.generation_cycle:
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S_new_u.add_vertex(V_new)
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else:
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S_new_u.add_vertex(v)
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S_new_u.add_vertex(s[-1]) # Second vertex on new long edge
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self.H[gen].append(S_new_l)
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self.H[gen].append(S_new_u)
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return
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# Plots
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def plot_complex(self):
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"""
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Here, C is the LIST of simplexes S in the
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2- or 3-D complex
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To plot a single simplex S in a set C, use e.g., [C[0]]
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"""
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from matplotlib import pyplot
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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:
|
||
|
|
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().__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().__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]
|