def lsq_pos_l1_penalty(matrix, rhs, cost_multiplier, weights_0):
"""
min 2-norm (matrix*w - rhs)** + 1-norm(cost_multiplier*(w-w0))
s.t. e'w = 1
w >= 0
"""
# define model
with Model('lsqSparse') as model:
# introduce n non-negative weight variables
weights = model.variable("weights", matrix.shape[1], Domain.inRange(0.0, +np.infty))
# e'*w = 1
model.constraint(Expr.sum(weights), Domain.equalsTo(1.0))
# sum of squared residuals
v = __l2_norm_squared(model, "2-norm(res)**", __residual(matrix, rhs, weights))
# \Gamma*(w - w0), p is an expression
p = Expr.mulElm(cost_multiplier, Expr.sub(weights, weights_0))
cost = model.variable("cost", matrix.shape[1], Domain.unbounded())
model.constraint(Expr.sub(cost, p), Domain.equalsTo(0.0))
t = __l1_norm(model, 'abs(weights)', cost)
# Minimise v + t
model.objective(ObjectiveSense.Minimize, __sum_weighted(1.0, v, 1.0, t))
# solve the problem
model.solve()
return np.array(weights.level())
def lsq_pos_l1_penalty(matrix, rhs, cost_multiplier, weights_0):
"""
min 2-norm (matrix*w - rhs)** + 1-norm(cost_multiplier*(w-w0))
s.t. e'w = 1
w >= 0
"""
# define model
with Model('lsqSparse') as model:
# introduce n non-negative weight variables
weights = model.variable("weights", matrix.shape[1],
Domain.inRange(0.0, +np.infty))
# e'*w = 1
model.constraint(Expr.sum(weights), Domain.equalsTo(1.0))
# sum of squared residuals
v = __l2_norm_squared(model, "2-norm(res)**",
__residual(matrix, rhs, weights))
# \Gamma*(w - w0), p is an expression
p = Expr.mulElm(cost_multiplier, Expr.sub(weights, weights_0))
cost = model.variable("cost", matrix.shape[1], Domain.unbounded())
model.constraint(Expr.sub(cost, p), Domain.equalsTo(0.0))
t = __l1_norm(model, 'abs(weights)', cost)
# Minimise v + t
model.objective(ObjectiveSense.Minimize,
__sum_weighted(1.0, v, 1.0, t))
# solve the problem
model.solve()
return np.array(weights.level())
def __linfty_norm(model, name, expr):
t = model.variable(name, 1, Domain.unbounded())
# (t, w_i) \in Q2
for i in range(0, expr.size()):
__quad_cone(model, t, expr.index(i))
return t
def __absolute(model, name, expr):
t = model.variable(name, expr.size(), Domain.unbounded())
# (t_i, w_i) \in Q2
for i in range(0, int(expr.size())):
__quad_cone(model, t.index(i), expr.index(i))
return t
def __linfty_norm(model, name, expr):
t = model.variable(name, 1, Domain.unbounded())
# (t, w_i) \in Q2
for i in range(0, expr.size()):
__quad_cone(model, t, expr.index(i))
return t
def __absolute(model, name, expr):
t = model.variable(name, expr.size(), Domain.unbounded())
# (t_i, w_i) \in Q2
for i in range(0, int(expr.size())):
__quad_cone(model, t.index(i), expr.index(i))
return t
def __l2_norm_squared(model, name, expr):
"""
Given an expression (e.g. a vector) this returns the squared L2-norm of this vector as an expression.
It also introduces an auxiliary variables. Mosek requires a name
for any variable that is added to a model. The user has to specify this name explicitly.
This requirement may disappear in future version of this API.
"""
t = model.variable(name, 1, Domain.unbounded())
__rotated_quad_cone(model, 0.5, t, expr)
return t
def __l2_norm_squared(model, name, expr):
"""
Given an expression (e.g. a vector) this returns the squared L2-norm of this vector as an expression.
It also introduces an auxiliary variables. Mosek requires a name
for any variable that is added to a model. The user has to specify this name explicitly.
This requirement may disappear in future version of this API.
"""
t = model.variable(name, 1, Domain.unbounded())
__rotated_quad_cone(model, 0.5, t, expr)
return t
def __Declare_SpeedUp_Vars(self, COModel):
n, N = self.n, 2 * self.m + 2 * self.n + len(self.roads)
if self.co_params['speedup']['Tau'] is True:
Tau = COModel.variable('Tau', 1, Domain.greaterThan(0.0)) # scalar
else:
Tau = COModel.variable('Tau', 1, Domain.unbounded()) # scalar
if self.co_params['speedup']['Eta'] is True:
Eta = COModel.variable('Eta', [n, n],
Domain.inPSDCone(n)) # n by n matrix
else:
Eta = COModel.variable('Eta', [n, n],
Domain.unbounded()) # n by n matrix
if self.co_params['speedup']['W'] is True:
W = COModel.variable('W', [N, N],
Domain.inPSDCone(N)) # N by N matrix
else:
W = COModel.variable('W', [N, N],
Domain.unbounded()) # N by N matrix
return Tau, Eta, W
def Build_Co_Model(self):
r = len(self.roads)
mu, sigma = self.mu, self.sigma
m, n, r = self.m, self.n, len(self.roads)
f, h = self.f, self.h
M, N = m + n + r, 2 * m + 2 * n + r
A = self.__Construct_A_Matrix()
A_Mat = Matrix.dense(A)
b = self.__Construct_b_vector()
# ---- build Mosek Model
COModel = Model()
# -- Decision Variable
Z = COModel.variable('Z', m, Domain.inRange(0.0, 1.0))
I = COModel.variable('I', m, Domain.greaterThan(0.0))
Alpha = COModel.variable('Alpha', M,
Domain.unbounded()) # M by 1 vector
Beta = COModel.variable('Beta', M, Domain.unbounded()) # M by 1 vector
Theta = COModel.variable('Theta', N,
Domain.unbounded()) # N by 1 vector
# M1_matrix related decision variables
'''
[tau, xi^T, phi^T
M1 = xi, eta, psi^t
phi, psi, w ]
'''
# no-need speedup variables
Psi = COModel.variable('Psi', [N, n], Domain.unbounded())
Xi = COModel.variable('Xi', n, Domain.unbounded()) # n by 1 vector
Phi = COModel.variable('Phi', N, Domain.unbounded()) # N by 1 vector
# has the potential to speedup
Tau, Eta, W = self.__Declare_SpeedUp_Vars(COModel)
# M2 matrix decision variables
'''
[a, b^T, c^T
M2 = b, e, d^t
c, d, f ]
'''
a_M2 = COModel.variable('a_M2', 1, Domain.greaterThan(0.0))
b_M2 = COModel.variable('b_M2', n, Domain.greaterThan(0.0))
c_M2 = COModel.variable('c_M2', N, Domain.greaterThan(0.0))
e_M2 = COModel.variable('e_M2', [n, n], Domain.greaterThan(0.0))
d_M2 = COModel.variable('d_M2', [N, n], Domain.greaterThan(0.0))
f_M2 = COModel.variable('f_M2', [N, N], Domain.greaterThan(0.0))
# -- Objective Function
obj_1 = Expr.dot(f, Z)
obj_2 = Expr.dot(h, I)
obj_3 = Expr.dot(b, Alpha)
obj_4 = Expr.dot(b, Beta)
obj_5 = Expr.dot([1], Expr.add(Tau, a_M2))
obj_6 = Expr.dot([2 * mean for mean in mu], Expr.add(Xi, b_M2))
obj_7 = Expr.dot(sigma, Expr.add(Eta, e_M2))
COModel.objective(
ObjectiveSense.Minimize,
Expr.add([obj_1, obj_2, obj_3, obj_4, obj_5, obj_6, obj_7]))
# Constraint 1
_expr = Expr.sub(Expr.mul(A_Mat.transpose(), Alpha), Theta)
_expr = Expr.sub(_expr, Expr.mul(2, Expr.add(Phi, c_M2)))
_expr_rhs = Expr.vstack(Expr.constTerm([0.0] * n), Expr.mul(-1, I),
Expr.constTerm([0.0] * M))
COModel.constraint('constr1', Expr.sub(_expr, _expr_rhs),
Domain.equalsTo(0.0))
del _expr, _expr_rhs
# Constraint 2
_first_term = Expr.add([
Expr.mul(Beta.index(row),
np.outer(A[row], A[row]).tolist()) for row in range(M)
])
_second_term = Expr.add([
Expr.mul(Theta.index(k), Matrix.sparse(N, N, [k], [k], [1]))
for k in range(N)
])
_third_term = Expr.add(W, f_M2)
_expr = Expr.sub(Expr.add(_first_term, _second_term), _third_term)
COModel.constraint('constr2', _expr, Domain.equalsTo(0.0))
del _expr, _first_term, _second_term, _third_term
# Constraint 3
_expr = Expr.mul(-2, Expr.add(Psi, d_M2))
_expr_rhs = Matrix.sparse([[Matrix.eye(n)], [Matrix.sparse(N - n, n)]])
COModel.constraint('constr3', Expr.sub(_expr, _expr_rhs),
Domain.equalsTo(0))
del _expr, _expr_rhs
# Constraint 4: I <= M*Z
COModel.constraint('constr4', Expr.sub(Expr.mul(20000.0, Z), I),
Domain.greaterThan(0.0))
# Constraint 5: M1 is SDP
COModel.constraint(
'constr5',
Expr.vstack(Expr.hstack(Tau, Xi.transpose(), Phi.transpose()),
Expr.hstack(Xi, Eta, Psi.transpose()),
Expr.hstack(Phi, Psi, W)), Domain.inPSDCone(1 + n + N))
return COModel
