def test_signed_curvature(self):
# Convex argument.
expr = cvx.abs(1 + cvx.exp(cvx.Variable()))
self.assertEqual(expr.curvature, s.CONVEX)
expr = cvx.abs(-cvx.entr(cvx.Variable()))
self.assertEqual(expr.curvature, s.UNKNOWN)
expr = cvx.abs(-cvx.log(cvx.Variable()))
self.assertEqual(expr.curvature, s.UNKNOWN)
# Concave argument.
expr = cvx.abs(cvx.log(cvx.Variable()))
self.assertEqual(expr.curvature, s.UNKNOWN)
expr = cvx.abs(-cvx.square(cvx.Variable()))
self.assertEqual(expr.curvature, s.CONVEX)
expr = cvx.abs(cvx.entr(cvx.Variable()))
self.assertEqual(expr.curvature, s.UNKNOWN)
# Affine argument.
expr = cvx.abs(cvx.NonNegative())
self.assertEqual(expr.curvature, s.CONVEX)
expr = cvx.abs(-cvx.NonNegative())
self.assertEqual(expr.curvature, s.CONVEX)
expr = cvx.abs(cvx.Variable())
self.assertEqual(expr.curvature, s.CONVEX)
def test_dcp_curvature(self):
expr = 1 + cvx.exp(cvx.Variable())
self.assertEqual(expr.curvature, s.CONVEX)
expr = cvx.Parameter() * cvx.NonNegative()
self.assertEqual(expr.curvature, s.AFFINE)
f = lambda x: x**2 + x**0.5
expr = f(cvx.Constant(2))
self.assertEqual(expr.curvature, s.CONSTANT)
expr = cvx.exp(cvx.Variable())**2
self.assertEqual(expr.curvature, s.CONVEX)
expr = 1 - cvx.sqrt(cvx.Variable())
self.assertEqual(expr.curvature, s.CONVEX)
expr = cvx.log(cvx.sqrt(cvx.Variable()))
self.assertEqual(expr.curvature, s.CONCAVE)
expr = -(cvx.exp(cvx.Variable()))**2
self.assertEqual(expr.curvature, s.CONCAVE)
expr = cvx.log(cvx.exp(cvx.Variable()))
self.assertEqual(expr.is_dcp(), False)
expr = cvx.entr(cvx.NonNegative())
self.assertEqual(expr.curvature, s.CONCAVE)
expr = ((cvx.Variable()**2)**0.5)**0
self.assertEqual(expr.curvature, s.CONSTANT)
def calculateEmbeddingMatrix(
D, verbose=False, target_dist=1.1, jlt_tries=30
):
assert isMetricSpaceMatrix(D)
n = D.shape[0]
if int(cvx.__version__.split(".")[0]) == 0:
Q = cvx.Semidef(n)
d = cvx.NonNegative()
else:
Q = cvx.Variable(shape=(n, n), PSD=True)
d = cvx.Variable(shape=(1, 1), nonneg=True)
# print(D)
# c = matrix(([1]*n))
log.debug("Generating constraints for embedding:")
log.debug(f"Distance matrix: {D}")
constraints = []
for i in range(n):
for j in range(i, n):
constraints += [D[i, j] ** 2 <= Q[i, i] + Q[j, j] - 2 * Q[i, j]]
constraints += [
Q[i, i] + Q[j, j] - 2 * Q[i, j] <= d * D[i, j] ** 2
]
log.debug(
f"adding constraint: {D[i,j]}**2 <= Q{[i,i]} + Q{[j,j]} - 2*Q{[i,j]}"
)
log.debug(
f"adding constraint: Q{[i,i]} + Q{[j,j]} - 2*Q{[i,j]} <= d * {D[i,j]}**2 "
)
obj = cvx.Minimize(d)
prob = cvx.Problem(obj, constraints)
solvers = cvx.installed_solvers()
if "MOSEK" in solvers:
log.info("Solving problem with MOSEK solver")
prob.solve(solver=cvx.MOSEK, verbose=verbose)
elif "CVXOPT" in solvers:
prob.solve(
solver=cvx.CVXOPT,
kktsolver=cvx.ROBUST_KKTSOLVER,
verbose=verbose,
)
log.info("Solving problem with CVXOPT solver")
else:
prob.solve(verbose=verbose)
log.warning(
"CVXOPT not installed. Solving problem with default solver."
)
if prob.status != cvx.OPTIMAL:
log.warning(
"embedding optimization status non-optimal: " + str(prob.status)
)
return None, None
# print(Q.value)
# print(np.linalg.eigvals(np.matrix(Q.value)))
# print(np.linalg.eigh(np.matrix(Q.value)))
# print(type(np.matrix(Q.value)))
# print(np.matrix(Q.value))
try:
L = np.linalg.cholesky(np.array(Q.value))
except np.linalg.LinAlgError:
eigenvals, eigenvecs = np.linalg.eigh(np.array(Q.value))
min_eigenv = min(eigenvals)
if min_eigenv < 0:
log.warning(
"Warning, matrix not positive semidefinite."
+ "Trying to correct for numerical errors with minimal eigenvalue: "
+ str(min_eigenv)
+ " (max. eigenvalue:"
+ str(max(eigenvals))
+ ")."
)
Q_new_t = (
np.transpose(eigenvecs) @ np.array(Q.value) @ eigenvecs
)
# print(eigenvals)
# print(Q_new_t) # should be = diagonal(eigenvalues)
# print(np.transpose(eigenvecs) * eigenvecs) # should be = Identity matrix
Q_new_t += np.diag([-min_eigenv] * len(eigenvals))
Q_new = eigenvecs @ Q_new_t @ np.transpose(eigenvecs)
L = np.linalg.cholesky(Q_new)
log.debug(f"Shape of lower-triangular matrix L: {L.shape}")
lowerdim, d = jlt_search(D, L, target_dist, num_tries=jlt_tries)
# print(lowerdim)
# return L,d.value
return lowerdim, d