def graph_implementation(var_args, size):
A = var_args[0]
n,m = A.size
# Requires that A is symmetric.
obj,constr = transpose.graph_implementation([A], (m,n))
constr += [AffEqConstraint(obj, A)]
t = Variable(*size).canonical_form()[0]
I = Constant(np.eye(n,m)).canonical_form()[0]
return (t, [SDP(A - I*t)] + constr)
def graph_implementation(var_args, size):
A = var_args[0] # m by n matrix.
n,m = A.size
# Create the equivalent problem:
# minimize (trace(U) + trace(V))/2
# subject to:
# [U A; A.T V] is positive semidefinite
X = Variable(n+m, n+m)
# Expand A.T.
obj,constr = transpose.graph_implementation([A], (m,n))
# Fix X using the fact that A must be affine by the DCP rules.
constr += [AffEqConstraint(X[0:n,n:n+m], A),
AffEqConstraint(X[n:n+m,0:n], obj)]
trace = 0.5*sum([X[i,i] for i in range(n+m)])
# Add SDP constraint.
return (trace, [SDP(X)] + constr)
def graph_implementation(var_args, size):
A = var_args[0] # m by n matrix.
n,m = A.size
# Create a matrix with Schur complement I*t - (1/t)*A.T*A.
X = Variable(n+m, n+m)
t = Variable().canonical_form()[0]
I_n = Constant(np.eye(n)).canonical_form()[0]
I_m = Constant(np.eye(m)).canonical_form()[0]
# Expand A.T.
obj,constr = transpose.graph_implementation([A], (m,n))
# Fix X using the fact that A must be affine by the DCP rules.
constr += [AffEqConstraint(X[0:n,0:n], I_n*t),
AffEqConstraint(X[0:n,n:n+m], A),
AffEqConstraint(X[n:n+m,0:n], obj),
AffEqConstraint(X[n:n+m,n:n+m], I_m*t),
]
# Add SDP constraint.
return (t, [SDP(X)] + constr)