def _asphericity_iteration(xk, l, m, p, A, a, B, b, verbose=0, solver='GLPK'):
# find better way to share all those parameters, maybe a class
#alphak = asphericity(xk)
R_xk = max(A.row(i) * vector(xk) + a[i] for i in range(m))
r_xk = min(B.row(j) * vector(xk) + b[j] for j in range(l))
alphak = R_xk / r_xk
a_LP = MixedIntegerLinearProgram(maximization=False, solver=solver)
x = a_LP.new_variable(integer=False, nonnegative=False)
z = a_LP.new_variable(integer=False, nonnegative=False)
for i in range(m):
for j in range(l):
aux = sum(
(A.row(i)[k] - alphak * B.row(j)[k]) * x[k] for k in range(p))
a_LP.add_constraint(aux + a[i] - alphak * b[j] <= z[0])
#solve LP
a_LP.set_objective(z[0])
if (verbose):
print '**** Solve LP ****'
a_LP.show()
opt_val = a_LP.solve()
x_opt = a_LP.get_values(x)
z_opt = a_LP.get_values(z)
if (verbose):
print 'Objective Value:', opt_val
for i, v in x_opt.iteritems():
print 'x_%s = %f' % (i, v)
for i, v in z_opt.iteritems():
print 'z = %f' % (v)
print '\n'
return [opt_val, x_opt, z_opt]
def support_function(P, d, verbose=0, return_xopt=False, solver='GLPK'):
r"""Compute support function of a convex polytope.
It is defined as `\max_{x in P} \langle x, d \rangle` , where `d` is an input vector.
INPUT:
* ``P`` - an object of class Polyhedron. It can also be given as ``[A, b]`` meaning `Ax \leq b`.
* ``d`` - a vector (or list) where the support function is evaluated.
* ``verbose`` - (default: 0) If 1, print the status of the LP.
* ``solver`` - (default: ``'GLPK'``) the LP solver used.
* ``return_xopt`` - (default: False) If True, the optimal point is returned, and ``sf = [oval, opt]``.
OUTPUT:
* ``sf`` - the value of the support function.
EXAMPLES::
sage: from polyhedron_tools.misc import BoxInfty, support_function
sage: P = BoxInfty([1,2,3], 1); P
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 8 vertices
sage: support_function(P, [1,1,1], return_xopt=True)
(9.0, {0: 2.0, 1: 3.0, 2: 4.0})
NOTES:
- The possibility of giving the input polytope as `[A, b]` instead of a
polyhedron is useful in cases when the dimensions are high (in practice,
above or around 20, but it depends on the particular system -number of
constraints- as well).
- If a different solver (e.g. ``guurobi``) is given, it should be installed properly.
"""
from sage.numerical.mip import MixedIntegerLinearProgram
if (type(P) == list):
A = P[0]
b = P[1]
elif P.is_empty():
# avoid formulating the LP if P = []
return 0
else: #assuming some form of Polyhedra
base_ring = P.base_ring()
# extract the constraints from P
m = len(P.Hrepresentation())
n = len(vector(P.Hrepresentation()[0])) - 1
b = vector(base_ring, m)
A = matrix(base_ring, m, n)
P_gen = P.Hrep_generator()
i = 0
for pigen in P_gen:
pi_vec = pigen.vector()
A.set_row(i, -pi_vec[1:len(pi_vec)])
b[i] = pi_vec[0]
i += 1
s_LP = MixedIntegerLinearProgram(maximization=True, solver=solver)
x = s_LP.new_variable(integer=False, nonnegative=False)
# objective function
obj = sum(d[i] * x[i] for i in range(len(d)))
s_LP.set_objective(obj)
s_LP.add_constraint(A * x <= b)
if (verbose):
print '**** Solve LP ****'
s_LP.show()
oval = s_LP.solve()
xopt = s_LP.get_values(x)
if (verbose):
print 'Objective Value:', oval
for i, v in xopt.iteritems():
print 'x_%s = %f' % (i, v)
print '\n'
if (return_xopt == True):
return oval, xopt
else:
return oval
