def interFacePolytope(a, b, LossMatrix):
N, M = LossMatrix.shape
# declare M ppl Variables
p = [ppl.Variable(j) for j in range(M)]
# declare polytope constraints
cs = ppl.Constraint_System()
# probabilies constraints on p
cs.insert(sum(p[j] for j in range(M)) == 1)
for j in range(M):
cs.insert(p[j] >= 0)
# strict Loss domination constraints for both a and b
Doma = scale_to_integers(domination_matrix(a, LossMatrix))
Domb = scale_to_integers(domination_matrix(b, LossMatrix))
for i in range(N):
if i != a:
# p is such that for any action i Loss[a,...]*p <= Loss[a,...]*p
cs.insert(sum((Doma[i, j] * p[j] for j in range(M))) <= 0)
if i != b:
# p is such that for any action i Loss[b,...]*p <= Loss[a,...]*p
cs.insert(sum((Domb[i, j] * p[j] for j in range(M))) <= 0)
return ppl.C_Polyhedron(cs)
def ppl_positive_cone(n):
r"""
Return the positive cone in R^n
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_positive_cone # optional - pplpy
sage: C = ppl_positive_cone(3) # optional - pplpy
sage: C.minimized_generators() # optional - pplpy
Generator_System {point(0/1, 0/1, 0/1), ray(0, 0, 1), ray(0, 1, 0), ray(1, 0, 0)}
"""
gs = ppl.Generator_System(ppl_zero_point(n))
for i in range(n):
gs.insert(ppl.ray(ppl.Variable(i)))
return ppl.C_Polyhedron(gs)
def rays(self):
"""
**Description:**
Returns the (not necessarily extremal) rays that generate the cone.
**Arguments:**
None.
**Returns:**
*(numpy.ndarray)* The list of rays that generate the cone.
**Example:**
We construct two cones and find their generating rays.
```python {3,6}
c1 = Cone([[0,1],[1,1]])
c2 = Cone(hyperplanes=[[0,1],[1,1]])
c1.rays()
# array([[0, 1],
# [1, 1]])
c2.rays()
# array([[ 1, 0],
# [-1, 1]])
```
"""
if self._ext_rays is not None:
return np.array(self._ext_rays)
if self._rays is not None:
return np.array(self._rays)
if (self._ambient_dim >= 12
and len(self._hyperplanes) != self._ambient_dim):
print("Warning: This operation might take a while for d > ~12 "
"and is likely impossible for d > ~18.")
cs = ppl.Constraint_System()
vrs = [ppl.Variable(i) for i in range(self._ambient_dim)]
for h in self.dual().extremal_rays():
cs.insert(
sum(h[i] * vrs[i] for i in range(self._ambient_dim)) >= 0)
cone = ppl.C_Polyhedron(cs)
rays = []
for gen in cone.minimized_generators():
if gen.is_ray():
rays.append(tuple(int(c) for c in gen.coefficients()))
elif gen.is_line():
rays.append(tuple(int(c) for c in gen.coefficients()))
rays.append(tuple(-int(c) for c in gen.coefficients()))
self._rays = np.array(rays, dtype=int)
self._dim = np.linalg.matrix_rank(self._rays)
return np.array(self._rays)
def ppl_cone(rays):
r"""
Convert the list ``rays`` into a ppl cone
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_cone # optional - pplpy
sage: C = ppl_cone([(0,1,2),(1,1,1),(1,0,1)]) # optional - pplpy
sage: C.minimized_generators() # optional - pplpy
Generator_System {point(0/1, 0/1, 0/1), ray(0, 1, 2), ray(1, 0, 1), ray(1, 1, 1)}
"""
n = len(rays[0])
gs = ppl.Generator_System(ppl_zero_point(n))
for r in rays:
gs.insert(ppl.ray(sum(j * ppl.Variable(i) for i, j in enumerate(r))))
return ppl.C_Polyhedron(gs)
def ppl_check_non_negative_cone(C):
r"""
Check whether the input cone ``C`` is a subcone of the positive cone
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_check_non_negative_cone # optional - pplpy
sage: import ppl # optional - pplpy
sage: gs = ppl.Generator_System() # optional - pplpy
sage: gs.insert(ppl.point()) # optional - pplpy
sage: gs.insert(ppl.ray(ppl.Variable(0) + ppl.Variable(2))) # optional - pplpy
sage: ppl_check_non_negative_cone(ppl.C_Polyhedron(gs)) # optional - pplpy
sage: gs.insert(ppl.ray(-ppl.Variable(0) - ppl.Variable(2))) # optional - pplpy
sage: ppl_check_non_negative_cone(ppl.C_Polyhedron(gs)) # optional - pplpy
Traceback (most recent call last):
...
ValueError: C must be a subcone of the non-negative cone
sage: gs = ppl.Generator_System() # optional - pplpy
sage: gs.insert(ppl.point(ppl.Variable(2))) # optional - pplpy
sage: ppl_check_non_negative_cone(ppl.C_Polyhedron(gs)) # optional - pplpy
Traceback (most recent call last):
...
ValueError: the cone does not contain zero
sage: gs.insert(ppl.point()) # optional - pplpy
sage: ppl_check_non_negative_cone(ppl.C_Polyhedron(gs)) # optional - pplpy
Traceback (most recent call last):
...
ValueError: should have only zero as vertices
"""
if not isinstance(C, ppl.C_Polyhedron):
raise ValueError('C must be a polyhedron in the right ambient space')
n = C.space_dimension()
if not C.contains(ppl.C_Polyhedron(ppl_zero_point(n))):
raise ValueError('the cone does not contain zero')
if not ppl_positive_cone(n).contains(C):
raise ValueError('C must be a subcone of the non-negative cone')
for g in C.generators():
if g.is_point() and not g.is_equivalent_to(ppl_zero_point(n)):
raise ValueError('should have only zero as vertices'.format(g))
if g.is_line():
raise ValueError('the cone contains a line')
def hyperplanes(self):
"""
**Description:**
Returns the inward-pointing normals to the hyperplanes that define the
cone.
**Arguments:**
None.
**Returns:**
*(numpy.ndarray)* The list of inward-pointing normals to the
hyperplanes that define the cone.
**Example:**
We construct two cones and find their hyperplane normals.
```python {3,6}
c1 = Cone([[0,1],[1,1]])
c2 = Cone(hyperplanes=[[0,1],[1,1]])
c1.hyperplanes()
# array([[ 1, 0],
# [-1, 1]])
c2.hyperplanes()
# array([[0, 1],
# [1, 1]])
```
"""
if self._hyperplanes is not None:
return np.array(self._hyperplanes)
if self._ambient_dim >= 12 and len(self.rays()) != self._ambient_dim:
print("Warning: This operation might take a while for d > ~12 "
"and is likely impossible for d > ~18.")
gs = ppl.Generator_System()
vrs = [ppl.Variable(i) for i in range(self._ambient_dim)]
gs.insert(ppl.point(0))
for r in self.extremal_rays():
gs.insert(
ppl.ray(sum(r[i] * vrs[i] for i in range(self._ambient_dim))))
cone = ppl.C_Polyhedron(gs)
hyperplanes = []
for cstr in cone.minimized_constraints():
hyperplanes.append(tuple(int(c) for c in cstr.coefficients()))
if cstr.is_equality():
hyperplanes.append(tuple(-int(c) for c in cstr.coefficients()))
self._hyperplanes = np.array(hyperplanes, dtype=int)
return np.array(self._hyperplanes)
def ppl_convert(P):
r"""
Convert a Sage polyhedron to a ppl polyhedron
EXAMPLES::
sage: from surface_dynamics.misc.ppl_utils import ppl_convert # optional - pplpy
sage: P = ppl_convert(Polyhedron(vertices=[(0,1,0),(1,0,1)], rays=[(0,0,1),[3,2,1]])) # optional - pplpy
sage: P.minimized_generators() # optional - pplpy
Generator_System {ray(0, 0, 1), point(0/1, 1/1, 0/1), point(1/1, 0/1, 1/1), ray(3, 2, 1)}
"""
if isinstance(P, ppl.C_Polyhedron):
return P
gs = ppl.Generator_System()
for v in P.vertices_list():
gs.insert(ppl.point(sum(j * ppl.Variable(i) for i, j in enumerate(v))))
for r in P.rays_list():
gs.insert(ppl.ray(sum(j * ppl.Variable(i) for i, j in enumerate(r))))
for l in P.lines_list():
gs.insert(ppl.line(sum(j * ppl.Variable(i) for i, j in enumerate(l))))
return ppl.C_Polyhedron(gs)
def DominationPolytope(i, LossMatrix):
N, M = LossMatrix.shape
# declare M ppl Variables
p = [ppl.Variable(j) for j in range(M)]
# declare polytope constraints
cs = ppl.Constraint_System()
# probabilies constraints on p
cs.insert(sum(p[j] for j in range(M)) == 1)
for j in range(M):
cs.insert(p[j] >= 0)
# strict Loss domination constraints
Dom = scale_to_integers(domination_matrix(i, LossMatrix))
for a in range(N):
if a != i:
# p is such that for any action a Loss[i,...]*p <= Loss[a,...]*p
#print "Domination line:", Dom[a,...], "inequality:", sum( (Dom[a,j]*p[j] for j in range(M)) ) <= 0
cs.insert(sum((Dom[a, j] * p[j] for j in range(M))) <= 0)
return ppl.C_Polyhedron(cs)
def parma_enumerate_vertices(A_ineq, b_ineq, A_eq=None, b_eq=None):
"""
Iterator over extreme points of polytope defined by linear equalities
and inequalities, A_ineq x <= b_ineq, A_eq x = b_eq.
This uses the Parma Polyhedral Library (PPL). Because PPL expects all
data to be rational, we enforce that inequalities and equalities are
specified by integer-valued matrices.
Parameters
----------
A_ineq : np.array
Inequalities matrix. Data type should be int,
shape should be (num_inequalities x num_variables)
b_ineq : np.array
Inequalities values. Data type should be int,
shape should be (num_inequalities)
A_eq : np.array
Equalities matrix. Data type should be int,
shape should be (num_equalities x num_variables)
b_eq : np.array
Equalities values. Data type should be int,
shape should be (num_equalities)
"""
try:
import ppl
except ImportError:
raise Exception("""
This method requires the Parma Polyhedra Library (PPL) library to
be installed. Normally, this can be done with
pip3 install cysignals gmpy2
pip3 install pplpy
Please see https://gitlab.com/videlec/pplpy for more information.
""")
def get_num_cons(A, b):
# Check data for validity and return number of constraints
if A is None:
assert (b is None or len(b) == 0)
num_cons = 0
else:
assert (isinstance(A, np.ndarray))
assert (isinstance(b, np.ndarray))
assert (np.issubdtype(A.dtype, np.integer))
assert (np.issubdtype(b.dtype, np.integer))
num_cons = A.shape[0]
assert (num_cons == len(b))
return num_cons
num_ineq_cons = get_num_cons(A_ineq, b_ineq)
num_eq_cons = get_num_cons(A_eq, b_eq)
if num_eq_cons == 0 and num_ineq_cons == 0:
raise Exception(
"Must specify at least one inequality or equality constrants")
if num_eq_cons > 0 and num_ineq_cons > 0:
assert (A_eq.shape[1] == A_ineq.shape[1])
num_vars = (A_eq if num_eq_cons > 0 else A_ineq).shape[1]
ppl_vars = [ppl.Variable(i) for i in range(num_vars)]
cs = ppl.Constraint_System()
for rowndx in range(num_ineq_cons):
if np.all(A_ineq[rowndx] == 0):
if b_ineq[rowndx] < 0:
raise Exception(
'Inequality constraint {} involves no variables and is unsatisfiable'
.format(rowndx))
else:
continue # trivial constraint
lhs = sum([
coef * ppl_vars[i] for i, coef in enumerate(A_ineq[rowndx])
if coef != 0
])
cs.insert(lhs <= b_ineq[rowndx])
for rowndx in range(num_eq_cons):
if np.all(A_eq[rowndx] == 0):
if b_eq[rowndx] != 0:
raise Exception(
'Equality constraint {} involves no variables and is unsatisfiable'
.format(rowndx))
else:
continue # trivial constraint
lhs = sum([
coef * ppl_vars[i] for i, coef in enumerate(A_eq[rowndx])
if coef != 0
])
cs.insert(lhs == b_eq[rowndx])
# convert linear inequalities into a list of extreme points
poly_from_constraints = ppl.C_Polyhedron(cs)
all_generators = poly_from_constraints.minimized_generators()
for p in all_generators:
if not p.is_point():
raise Exception(
'Returned solution not a point: {}. '.format(p) +
'Typically this means that linear constraints specify a cone, not a polytope'
)
# Convert a solution vector in ppl format to a numpy array
x = np.fromiter(p.coefficients(), dtype='double')
x = x / float(p.divisor())
yield x
def is_solid(self, backend=None, c=0.01, verbose=0):
"""
**Description:**
Returns True if the cone is solid, i.e. if it is full-dimensional.
:::note
If the generating rays are known then this can simply be checked by
computing the dimension of the linear space that they span. However,
when only the hyperplane inequalities are known this can be a difficult
problem. When using PPL as the backend, the convex hull is explicitly
constructed and checked. The other backends try to solve an
optimization problem inside the stretched cone, which fails if the cone
is not solid. The latter approach is much faster, but there can be
extremely narrow cones where the optimization fails and this function
returns a false negative. Mosek is recommended when using such
extremely narrow cones.
:::
**Arguments:**
- `backend` *(str, optional)*: Specifies which backend to use.
Available options are "ppl", "ortools", and any backends available
for the [`tip_of_stretched_cone`](#tip_of_stretched_cone)
function. If not specified, it tries all available backends.
- `c` *(float, optional, default=0.01)*: A number used to create
the stretched cone and try to find the tip. This is ignored when
using PPL as the backend.
- `verbose` *(int, optional, default=0)*: The verbosity level.
- verbose = 0: Do not print anything.
- verbose = 1: Print warnings when optimizers fail.
**Returns:**
*(bool)* The truth value of the cone being solid.
**Example:**
We construct two cones and check if they are solid.
```python {3,5}
c1 = Cone([[1,0],[0,1]])
c2 = Cone([[1,0,0],[0,1,0]])
c1.is_solid()
# True
c2.is_solid()
# False
```
"""
if self._is_solid is not None:
return self._is_solid
if self._rays is not None:
return np.linalg.matrix_rank(self._rays) == self._ambient_dim
if backend is None:
backend = ("mosek" if config.mosek_is_activated() else "ortools")
if backend == "ppl":
cs = ppl.Constraint_System()
vrs = [ppl.Variable(i) for i in range(self._ambient_dim)]
for h in self._hyperplanes:
cs.insert(
sum(h[i] * vrs[i] for i in range(self._ambient_dim)) >= 0)
cone = ppl.C_Polyhedron(cs)
self._is_solid = cone.affine_dimension() == self._ambient_dim
return self._is_solid
if backend == "ortools":
hp = self._hyperplanes.tolist()
solve_ctr = 0
while solve_ctr < 10:
obj_vec = np.dot(np.random.random(size=len(hp)), hp)
solver = pywraplp.Solver(
"find_pt", pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)
obj = solver.Objective()
var = []
for i in range(self._ambient_dim):
var.append(
solver.NumVar(-solver.infinity(), solver.infinity(),
f"x_{i}"))
obj.SetCoefficient(var[-1], obj_vec[i])
obj.SetMinimization()
cons_list = []
for v in hp:
cons_list.append(solver.Constraint(c, solver.infinity()))
for j in range(self._ambient_dim):
cons_list[-1].SetCoefficient(var[j], v[j])
status = solver.Solve()
if status in (solver.FEASIBLE, solver.OPTIMAL):
self._is_solid = True
return self._is_solid
elif status == solver.INFEASIBLE:
self._is_solid = False
return self._is_solid
else:
if verbose >= 1:
print(
f"Solver returned status: {status}. Trying again.")
solve_ctr += 1
if verbose >= 1:
print("Linear solver failed too many times. "
"Assuming problem infeasible.")
self._is_solid = False
return self._is_solid
if backend in ("all", "mosek", "cvxopt"):
opt_res = None
try:
opt_res = self.tip_of_stretched_cone(c,
backend=backend,
verbose=verbose)
except:
pass
self._is_solid = opt_res is not None
return self._is_solid
else:
backends = ["ppl", "ortools", "all", "mosek", "cvxopt"]
raise Exception(f"Available options for backends are: {backends}")
