def solve_sdp_program(W):
assert W.ndim == 2
assert W.shape[0] == W.shape[1]
W = W.copy()
n = W.shape[0]
W = expand_matrix(W)
with Model('gw_max_3_cut') as M:
W = Matrix.dense(W / 3.)
J = Matrix.ones(3*n, 3*n)
# variable
Y = M.variable('Y', Domain.inPSDCone(3*n))
# objective function
M.objective(ObjectiveSense.Maximize, Expr.dot(W, Expr.sub(J, Y)))
# constraints
for i in range(3*n):
M.constraint(f'c_{i}{i}', Y.index(i, i), Domain.equalsTo(1.))
for i in range(n):
M.constraint(f'c_{i}^01', Y.index(i*3, i*3+1), Domain.equalsTo(-1/2.))
M.constraint(f'c_{i}^02', Y.index(i*3, i*3+2), Domain.equalsTo(-1/2.))
M.constraint(f'c_{i}^12', Y.index(i*3+1, i*3+2), Domain.equalsTo(-1/2.))
for j in range(i+1, n):
for a, b in product(range(3), repeat=2):
M.constraint(f'c_{i}{j}^{a}{b}-0', Y.index(i*3 + a, j*3 + b), Domain.greaterThan(-1/2.))
M.constraint(f'c_{i}{j}^{a}{b}-1', Expr.sub(Y.index(i*3 + a, j*3 + b), Y.index(i*3 + (a + 1) % 3, j*3 + (b + 1) % 3)), Domain.equalsTo(0.))
M.constraint(f'c_{i}{j}^{a}{b}-2', Expr.sub(Y.index(i*3 + a, j*3 + b), Y.index(i*3 + (a + 2) % 3, j*3 + (b + 2) % 3)), Domain.equalsTo(0.))
# solution
M.solve()
Y_opt = Y.level()
return np.reshape(Y_opt, (3*n,3*n))
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 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
model = mModel.build_model('lsqSparse')
# introduce n non-negative weight variables
weights = mModel.weights(model, "weights", n=matrix.shape[1], lb=0.0)
# e'*w = 1
mBound.equal(model, Expr.sum(weights), 1.0)
# sum of squared residuals
v = mMath.l2_norm_squared(model, "2-norm(res)**", __residual(matrix, rhs, weights))
print matrix.shape[1]
print weights_0
print weights
# \Gamma*(w - w0), p is an expression
p = mMath.mat_vec_prod(cost_multiplier, Expr.sub(weights, weights_0))
t = mMath.l1_norm(model, 'abs(weights)', p)
# Minimise v + t
mModel.minimise(model, __sum_weighted(1.0, v, 1.0, t))
return np.array(weights.level())
def _update(self, clique_data, parent=None):
for c in clique_data['cliques']:
self.cliques[c['name']] = v = self.model.variable(
c['name'],
Domain.inRange(0.0, 1.0),
Domain.isInteger()
)
for img, cmd in product(c['images'], c['commands']):
self.by_img_cmd[img,cmd].append(v)
self._obj.append(Expr.mul(float(c['time']), v))
if parent is not None:
self.model.constraint(
'c-%s-%s' % (c['name'], parent['name']),
Expr.sub(v, self.cliques[parent['name']]),
Domain.lessThan(0.0)
)
for child in c['children']:
self._update(child, c)
for inter in clique_data['intersections']:
self.model.constraint(
'iter-%d' % self._inter,
Expr.add([self.cliques[i] for i in inter]),
Domain.lessThan(1.0)
)
self._inter += 1
def solve_sdp_program(A):
assert A.ndim == 2
assert A.shape[0] == A.shape[1]
A = A.copy()
n = A.shape[0]
with Model('theta_2') as M:
# variable
X = M.variable('X', Domain.inPSDCone(n + 1))
t = M.variable()
# objective function
M.objective(ObjectiveSense.Maximize, t)
# constraints
for i in range(n + 1):
M.constraint(f'c{i}{i}', X.index(i, i), Domain.equalsTo(1.))
if i == 0:
continue
M.constraint(f'c0,{i}', Expr.sub(X.index(0, i), t),
Domain.greaterThan(0.))
for j in range(i + 1, n + 1):
if A[i - 1, j - 1] == 0:
M.constraint(f'c{i},{j}', X.index(i, j),
Domain.equalsTo(0.))
# solution
M.solve()
X_sol = X.level()
t_sol = t.level()
t_sol = t_sol[0]
theta = 1. / t_sol**2
return theta
def Markowitz(mu, covar, alpha, L=1.0):
# defineerime kasutatava mudeli kasutades keskväärtusi
model = mosek_model.build_model('meanVariance')
# toome sisse n mitte-negatiivset kaalu
weights = mosek_model.weights(model, "weights", n=covar.shape[1])
# leiame kaaludele vastavad finantsvõimendused
lev = mosek_math.l1_norm(model, "leverage", weights)
# nõuame, et oleks täidetud tingimus omega_1 + ... + omega_n = 1
mosek_bound.equal(model, Expr.sum(weights), 1.0)
# nõuame, et oleks täidetud tingimus |omega_1| + ... + |omega_n| <= L
mosek_bound.upper(model, lev, L)
v = Expr.dot(mu.values, weights)
# arvutame variatsiooni
var = mosek_math.variance(model, "variance", weights, alpha*covar.values)
# maksimeerime sihifunktsiooni
mosek_model.maximise(model, Expr.sub(v, var))
# arvutame lõpuks kaalud ja tagastame need
weights = pd.Series(data=np.array(weights.level()), index=stocks.keys())
return weights
def solve(x0,
risk_alphas,
loadings,
srisk,
cost_per_trade=DEFAULT_COST,
max_risk=0.01):
N = len(x0)
# don't hold no risk data (likely dead)
lim = np.where(srisk.isnull(), 0.0, 1.0)
loadings = loadings.fillna(0)
srisk = srisk.fillna(0)
risk_alphas = risk_alphas.fillna(0)
with Model() as m:
w = m.variable(N, Domain.inRange(-lim, lim))
longs = m.variable(N, Domain.greaterThan(0))
shorts = m.variable(N, Domain.greaterThan(0))
gross = m.variable(N, Domain.greaterThan(0))
m.constraint(
"leverage_consistent",
Expr.sub(gross, Expr.add(longs, shorts)),
Domain.equalsTo(0),
)
m.constraint("net_consistent", Expr.sub(w, Expr.sub(longs, shorts)),
Domain.equalsTo(0.0))
m.constraint("leverage_long", Expr.sum(longs), Domain.lessThan(1.0))
m.constraint("leverage_short", Expr.sum(shorts), Domain.lessThan(1.0))
buys = m.variable(N, Domain.greaterThan(0))
sells = m.variable(N, Domain.greaterThan(0))
gross_trade = Expr.add(buys, sells)
net_trade = Expr.sub(buys, sells)
total_gross_trade = Expr.sum(gross_trade)
m.constraint(
"net_trade",
Expr.sub(w, net_trade),
Domain.equalsTo(np.asarray(x0)), # cannot handle series
)
# add risk constraint
vol = m.variable(1, Domain.lessThan(max_risk))
stacked = Expr.vstack(vol.asExpr(), Expr.mulElm(w, srisk.values))
stacked = Expr.vstack(stacked, Expr.mul(loadings.values.T, w))
m.constraint("vol-cons", stacked, Domain.inQCone())
alphas = risk_alphas.dot(np.vstack([loadings.T, np.diag(srisk)]))
gain = Expr.dot(alphas, net_trade)
loss = Expr.mul(cost_per_trade, total_gross_trade)
m.objective(ObjectiveSense.Maximize, Expr.sub(gain, loss))
m.solve()
result = pd.Series(w.level(), srisk.index)
return result
def markowitz(exp_ret, covariance_mat, aversion):
# define model
model = mModel.build_model("mean_var")
# set of n weights (unconstrained)
weights = mModel.weights(model, "weights", n=len(exp_ret))
mBound.equal(model, Expr.sum(weights), 1.0)
# standard deviation induced by covariance matrix
var = mMath.variance(model, "var", weights, covariance_mat)
mModel.maximise(model=model, expr=Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var)))
return np.array(weights.level())
def markowitz(exp_ret, covariance_mat, aversion):
# define model
with Model("mean var") as model:
# set of n weights (unconstrained)
weights = model.variable("weights", len(exp_ret), Domain.inRange(-np.infty, +np.infty))
model.constraint(Expr.sum(weights), Domain.equalsTo(1.0))
# standard deviation induced by covariance matrix
var = __variance(model, "var", weights, covariance_mat)
model.objective(ObjectiveSense.Maximize, Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var)))
model.solve()
#mModel.maximise(model=model, expr=Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var)))
return np.array(weights.level())
def markowitz(exp_ret, covariance_mat, aversion):
# define model
with Model("mean var") as model:
# set of n weights (unconstrained)
weights = model.variable("weights", len(exp_ret),
Domain.inRange(-np.infty, +np.infty))
model.constraint(Expr.sum(weights), Domain.equalsTo(1.0))
# standard deviation induced by covariance matrix
var = __variance(model, "var", weights, covariance_mat)
model.objective(
ObjectiveSense.Maximize,
Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var)))
model.solve()
#mModel.maximise(model=model, expr=Expr.sub(Expr.dot(exp_ret, weights), Expr.mul(aversion, var)))
return np.array(weights.level())
def solve_sdp_program(W):
assert W.ndim == 2
assert W.shape[0] == W.shape[1]
W = W.copy()
n = W.shape[0]
with Model('gw_max_cut') as M:
W = Matrix.dense(W / 4.)
J = Matrix.ones(n, n)
# variable
Y = M.variable('Y', Domain.inPSDCone(n))
# objective function
M.objective(ObjectiveSense.Maximize, Expr.dot(W, Expr.sub(J, Y)))
# constraints
for i in range(n):
M.constraint(f'c_{i}', Y.index(i, i), Domain.equalsTo(1.))
# solve
M.solve()
# solution
Y_opt = Y.level()
return np.reshape(Y_opt, (n, n))
def solve_sdp_program(W, k):
assert W.ndim == 2
assert W.shape[0] == W.shape[1]
W = W.copy()
n = W.shape[0]
with Model('fj_max_k_cut') as M:
W = Matrix.dense((k - 1) / (2 * k) * W)
J = Matrix.ones(n, n)
# variable
Y = M.variable('Y', Domain.inPSDCone(n))
# objective function
M.objective(ObjectiveSense.Maximize, Expr.dot(W, Expr.sub(J, Y)))
# constraints
for i in range(n):
M.constraint(f'c_{i}', Y.index(i, i), Domain.equalsTo(1.))
for j in range(i + 1, n):
M.constraint(f'c_{i},{j}', Y.index(i, j),
Domain.greaterThan(-1 / (k - 1)))
# solution
M.solve()
Y_opt = Y.level()
return np.reshape(Y_opt, (n, n))
def _update(self, clique_data, parent=None):
for c in clique_data['cliques']:
self.cliques[c['name']] = v = self.model.variable(
c['name'], Domain.inRange(0.0, 1.0), Domain.isInteger())
for img, cmd in product(c['images'], c['commands']):
self.by_img_cmd[img, cmd].append(v)
self._obj.append(Expr.mul(float(c['time']), v))
if parent is not None:
self.model.constraint(
'c-%s-%s' % (c['name'], parent['name']),
Expr.sub(v, self.cliques[parent['name']]),
Domain.lessThan(0.0))
for child in c['children']:
self._update(child, c)
for inter in clique_data['intersections']:
self.model.constraint('iter-%d' % self._inter,
Expr.add([self.cliques[i] for i in inter]),
Domain.lessThan(1.0))
self._inter += 1
q_2_b = m.variable('q_2_b', *binary)
q_2_c = m.variable('q_2_c', *binary)
q_2_d = m.variable('q_2_d', *binary)
q_3_b = m.variable('q_3_b', *binary)
q_3_c = m.variable('q_3_c', *binary)
q_3_d = m.variable('q_3_d', *binary)
# Inclusion of an image and a command means that image must
# use all command invocation from the clique.
# For instance:
# (1) z_1_a <= w_1
# (2) z_1_a <= y_a
# (3) z_1_a >= w_1 + y_a - 1
m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0))
m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0))
m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])), Domain.greaterThan(-1.0))
m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0))
m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0))
m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])), Domain.greaterThan(-1.0))
m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])), Domain.greaterThan(-1.0))
m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])), Domain.greaterThan(-1.0))
m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0))
m.constraint('c_2_a_2', Expr.sub(z_2_a, y_a), Domain.lessThan(0.0))
m.constraint('c_2_a_3', Expr.sub(z_2_a, Expr.add([w_2, y_a])), Domain.greaterThan(-1.0))
m.constraint('c_2_b_1', Expr.sub(z_2_b, w_2), Domain.lessThan(0.0))
def solve(self, problem, saver):
# Construct model.
self.problem = problem
self.model = model = Model()
if self.time is not None:
model.setSolverParam('mioMaxTime', 60.0 * int(self.time))
# x[1,c] = 1 if the master schedule has (null, c) in its first stage
# x[s,c1,c2] = 1 if the master schedule has (c1, c2) in stage s > 1
x = {}
for s in problem.all_stages:
if s == 1:
# First arc in the individual image path.
for c in problem.commands:
x[1,c] = model.variable(
'x[1,%s]' % c, 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger()
)
else:
# Other arcs.
for c1, c2 in product(problem.commands, problem.commands):
if c1 == c2:
continue
x[s,c1,c2] = model.variable(
'x[%s,%s,%s]' % (s,c1,c2), 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger()
)
smax = max(problem.all_stages)
obj = [0.0]
# TODO: deal with images that do not have the same number of commands.
# t[s,c] is the total time incurred at command c in stage s
t = {}
for s in problem.all_stages:
for c in problem.commands:
t[s,c] = model.variable(
't[%s,%s]' % (c,s), 1,
Domain.greaterThan(0.0)
)
if s == 1:
model.constraint('t[1,%s]' % c,
Expr.sub(t[1,c], Expr.mul(float(problem.commands[c]), x[1,c])),
Domain.greaterThan(0.0)
)
else:
rhs = [0.0]
for c1, coeff in problem.commands.items():
if c1 == c:
continue
else:
rhs = Expr.add(rhs, Expr.mulElm(t[s-1,c1], x[s,c1,c]))
model.constraint('t[%s,%s]' % (s,c),
Expr.sub(t[1,c], rhs),
Domain.greaterThan(0.0)
)
# Objective function = sum of aggregate comand times
if s == smax:
obj = Expr.add(obj, t[s,c])
# y[i,1,c] = 1 if image i starts by going to c
# y[i,s,c1,c2] = 1 if image i goes from command c1 to c2 in stage s > 1
y = {}
for i, cmds in problem.images.items():
for s in problem.stages[i]:
if s == 1:
# First arc in the individual image path.
for c in cmds:
y[i,1,c] = model.variable(
'y[%s,1,%s]' % (i,c), 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger()
)
model.constraint('x_y[i%s,1,c%s]' % (i,c),
Expr.sub(x[1,c], y[i,1,c]),
Domain.greaterThan(0.0)
)
else:
# Other arcs.
for c1, c2 in product(cmds, cmds):
if c1 == c2:
continue
y[i,s,c1,c2] = model.variable(
'y[%s,%s,%s,%s]' % (i,s,c1,c2), 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger()
)
model.constraint('x_y[i%s,s%s,c%s,c%s]' % (i,s,c1,c2),
Expr.sub(x[s,c1,c2], y[i,s,c1,c2]),
Domain.greaterThan(0.0)
)
for c in cmds:
# Each command is an arc destination exactly once.
arcs = [y[i,1,c]]
for c1 in cmds:
if c1 == c:
continue
arcs.extend([y[i,s,c1,c] for s in problem.stages[i][1:]])
model.constraint('y[i%s,c%s]' % (i,c),
Expr.add(arcs),
Domain.equalsTo(1.0)
)
# Network balance equations (stages 2 to |stages|-1).
# Sum of arcs in = sum of arcs out.
for s in problem.stages[i][:len(problem.stages[i])-1]:
if s == 1:
arcs_in = [y[i,1,c]]
else:
arcs_in = [y[i,s,c1,c] for c1 in cmds if c1 != c]
arcs_out = [y[i,s+1,c,c2] for c2 in cmds if c2 != c]
model.constraint('y[i%s,s%s,c%s]' % (i,s,c),
Expr.sub(Expr.add(arcs_in), Expr.add(arcs_out)),
Domain.equalsTo(0.0)
)
model.objective('z', ObjectiveSense.Minimize, Expr.add(x.values()))
# model.objective('z', ObjectiveSense.Minimize, obj)
model.setLogHandler(sys.stdout)
model.acceptedSolutionStatus(AccSolutionStatus.Feasible)
model.solve()
# Create optimal schedule.
schedule = defaultdict(list)
for i, cmds in problem.images.items():
for s in problem.stages[i]:
if s == 1:
# First stage starts our walk.
for c in cmds:
if y[i,s,c].level()[0] > 0.5:
schedule[i].append(c)
break
else:
# After that we know what our starting point is.
for c2 in cmds:
if c2 == c:
continue
if y[i,s,c,c2].level()[0] > 0.5:
schedule[i].append(c2)
c = c2
break
saver(schedule)
z_2_a = m.variable('z_2_a', *binary)
z_2_b = m.variable('z_2_b', *binary)
z_2_c = m.variable('z_2_c', *binary)
z_2_d = m.variable('z_2_d', *binary)
z_3_b = m.variable('z_3_b', *binary)
z_3_c = m.variable('z_3_c', *binary)
z_3_d = m.variable('z_3_d', *binary)
# Inclusion of an image and a command means that image must
# use all command invocation from the clique.
# For instance:
# (1) z_1_a <= w_1
# (2) z_1_a <= y_a
# (3) z_1_a >= w_1 + y_a - 1
m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0))
m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0))
m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])),
Domain.greaterThan(-1.0))
m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0))
m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0))
m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])),
Domain.greaterThan(-1.0))
m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])),
Domain.greaterThan(-1.0))
m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])),
Domain.greaterThan(-1.0))
m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0))
m.constraint('c_2_a', Expr.add([x_2_a, x_12_a]), Domain.equalsTo(1.0))
m.constraint('c_2_b', Expr.add([x_2_b, x_23_bcd, x_123_b]),
Domain.equalsTo(1.0))
m.constraint('c_2_c', Expr.add([x_2_c, x_23_bcd, x_123_b_23_cd]),
Domain.equalsTo(1.0))
m.constraint('c_2_d', Expr.add([x_2_d, x_23_bcd, x_123_b_23_cd]),
Domain.equalsTo(1.0))
m.constraint('c_3_b', Expr.add([x_3_b, x_23_bcd, x_123_b]),
Domain.equalsTo(1.0))
m.constraint('c_3_c', Expr.add([x_3_c, x_23_bcd, x_123_b_23_cd]),
Domain.equalsTo(1.0))
m.constraint('c_3_d', Expr.add([x_3_d, x_23_bcd, x_123_b_23_cd]),
Domain.equalsTo(1.0))
# Add dependency constraints for sub-cliques.
m.constraint('d_123_b_23_cd', Expr.sub(x_123_b, x_123_b_23_cd),
Domain.greaterThan(0.0))
# Eliminated intersections between cliques.
m.constraint('e1', Expr.add([x_23_bcd, x_123_b]), Domain.lessThan(1.0))
m.constraint('e2', Expr.add([x_12_a, x_123_b]), Domain.lessThan(1.0))
m.constraint('e3', Expr.add([x_12_a, x_23_bcd]), Domain.lessThan(1.0))
# Minimize resources required to construct all images.
obj = [
Expr.mul(c, x) for c, x in [
# Individual image/command pairs
(r['A'], x_1_a),
(r['B'], x_1_b),
(r['A'], x_2_a),
(r['B'], x_2_b),
x_123_b = m.variable('x_123_b', *binary)
x_123_b_23_cd = m.variable('x_123_b_23_cd', *binary)
# Each command must be run once for each image.
m.constraint('c_1_a', Expr.add([x_1_a]), Domain.equalsTo(1.0))
m.constraint('c_1_b', Expr.add([x_1_b, x_123_b]), Domain.equalsTo(1.0))
m.constraint('c_2_a', Expr.add([x_2_a]), Domain.equalsTo(1.0))
m.constraint('c_2_b', Expr.add([x_2_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0))
m.constraint('c_2_c', Expr.add([x_2_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
m.constraint('c_2_d', Expr.add([x_2_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
m.constraint('c_3_b', Expr.add([x_3_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0))
m.constraint('c_3_c', Expr.add([x_3_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
m.constraint('c_3_d', Expr.add([x_3_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
# Add dependency constraints for sub-cliques.
m.constraint('d_123_b_23_cd', Expr.sub(x_123_b, x_123_b_23_cd), Domain.greaterThan(0.0))
# Eliminated intersections between cliques.
m.constraint('e1', Expr.add([x_23_bcd, x_123_b]), Domain.lessThan(1.0))
# Minimize resources required to construct all images.
obj = [Expr.mul(c, x) for c, x in [
# Individual image/command pairs
(r['A'], x_1_a), (r['B'], x_1_b),
(r['A'], x_2_a), (r['B'], x_2_b), (r['C'], x_2_c), (r['D'], x_2_d),
(r['B'], x_3_b), (r['C'], x_3_c), (r['D'], x_3_d),
# Cliques
(r['B'] + r['C'] + r['D'], x_23_bcd),
(r['B'], x_123_b),
(r['C'] + r['D'], x_123_b_23_cd),
# clique or incur its own cost.
q_2_b = m.variable('q_2_b', *binary)
q_2_c = m.variable('q_2_c', *binary)
q_2_d = m.variable('q_2_d', *binary)
q_3_b = m.variable('q_3_b', *binary)
q_3_c = m.variable('q_3_c', *binary)
q_3_d = m.variable('q_3_d', *binary)
# Inclusion of an image and a command means that image must
# use all command invocation from the clique.
# For instance:
# (1) z_1_a <= w_1
# (2) z_1_a <= y_a
# (3) z_1_a >= w_1 + y_a - 1
m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0))
m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0))
m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])), Domain.greaterThan(-1.0))
m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0))
m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0))
m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])), Domain.greaterThan(-1.0))
m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])), Domain.greaterThan(-1.0))
m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])), Domain.greaterThan(-1.0))
m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0))
m.constraint('c_2_a_2', Expr.sub(z_2_a, y_a), Domain.lessThan(0.0))
m.constraint('c_2_a_3', Expr.sub(z_2_a, Expr.add([w_2, y_a])), Domain.greaterThan(-1.0))
m.constraint('c_2_b_1', Expr.sub(z_2_b, w_2), Domain.lessThan(0.0))
x_12_a = m.variable('x_12_a', *binary)
# Each command must be run once for each image.
m.constraint('c_1_a', Expr.add([x_1_a, x_12_a]), Domain.equalsTo(1.0))
m.constraint('c_1_b', Expr.add([x_1_b, x_123_b]), Domain.equalsTo(1.0))
m.constraint('c_2_a', Expr.add([x_2_a, x_12_a]), Domain.equalsTo(1.0))
m.constraint('c_2_b', Expr.add([x_2_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0))
m.constraint('c_2_c', Expr.add([x_2_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
m.constraint('c_2_d', Expr.add([x_2_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
m.constraint('c_3_b', Expr.add([x_3_b, x_23_bcd, x_123_b]), Domain.equalsTo(1.0))
m.constraint('c_3_c', Expr.add([x_3_c, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
m.constraint('c_3_d', Expr.add([x_3_d, x_23_bcd, x_123_b_23_cd]), Domain.equalsTo(1.0))
# Add dependency constraints for sub-cliques.
m.constraint('d_123_b_23_cd', Expr.sub(x_123_b, x_123_b_23_cd), Domain.greaterThan(0.0))
# Eliminated intersections between cliques.
m.constraint('e1', Expr.add([x_23_bcd, x_123_b]), Domain.lessThan(1.0))
m.constraint('e2', Expr.add([x_12_a, x_123_b]), Domain.lessThan(1.0))
m.constraint('e3', Expr.add([x_12_a, x_23_bcd]), Domain.lessThan(1.0))
# Minimize resources required to construct all images.
obj = [Expr.mul(c, x) for c, x in [
# Individual image/command pairs
(r['A'], x_1_a), (r['B'], x_1_b),
(r['A'], x_2_a), (r['B'], x_2_b), (r['C'], x_2_c), (r['D'], x_2_d),
(r['B'], x_3_b), (r['C'], x_3_c), (r['D'], x_3_d),
# Cliques
(r['B'] + r['C'] + r['D'], x_23_bcd),
def solve(self, problem, saver):
# Construct model.
self.problem = problem
self.model = model = Model()
if self.time is not None:
model.setSolverParam('mioMaxTime', 60.0 * int(self.time))
# x[i,s,c] = 1 if image i runs command c during stage s, 0 otherwise.
self.x = x = {}
for i, cmds in problem.images.items():
for s, c in product(problem.stages[i], cmds):
x[i,s,c] = model.variable(
'x[%s,%s,%s]' % (i,s,c), 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger()
)
# y[ip,iq,s,c] = 1 if images ip & iq have a shared path through stage
# s by running command c during s, 0 otherwise.
y = {}
for (ip, iq), cmds in problem.shared_cmds.items():
for s, c in product(problem.shared_stages[ip, iq], cmds):
y[ip,iq,s,c] = model.variable(
'y[%s,%s,%s,%s]' % (ip,iq,s,c), 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger()
# Domain.inRange(0.0, 1.0)
)
# TODO: need to remove presolved commands so the heuristic doesn't try them.
# TODO: Add a heuristic initial solution.
# TODO: Presolving
# Each image one command per stage, and each command once.
for i in problem.images:
for s in problem.stages[i]:
model.constraint('c1[%s,%s]' % (i,s),
Expr.add([x[i,s,c] for c in problem.images[i]]),
Domain.equalsTo(1.0)
)
for c in problem.images[i]:
model.constraint('c2[%s,%s]' % (i,c),
Expr.add([x[i,s,c] for s in problem.stages[i]]),
Domain.equalsTo(1.0)
)
# Find shared paths among image pairs.
for (ip, iq), cmds in problem.shared_cmds.items():
for s in problem.shared_stages[ip,iq]:
for c in cmds:
model.constraint('c3[%s,%s,%s,%s]' % (ip,iq,s,c),
Expr.sub(y[ip,iq,s,c], x[ip,s,c]),
Domain.lessThan(0.0)
)
model.constraint('c4[%s,%s,%s,%s]' % (ip,iq,s,c),
Expr.sub(y[ip,iq,s,c], x[iq,s,c]),
Domain.lessThan(0.0)
)
if s > 1:
lhs = Expr.add([y[ip,iq,s,c] for c in cmds])
rhs = Expr.add([y[ip,iq,s-1,c] for c in cmds])
model.constraint('c5[%s,%s,%s,%s]' % (ip,iq,s,c),
Expr.sub(lhs, rhs), Domain.lessThan(0.0)
)
if y:
obj = Expr.add(y.values())
else:
obj = 0.0
model.objective('z', ObjectiveSense.Maximize, obj)
model.setLogHandler(sys.stdout)
model.acceptedSolutionStatus(AccSolutionStatus.Feasible)
model.solve()
# Create optimal schedule.
schedule = defaultdict(list)
for i, stages in problem.stages.items():
for s in stages:
for c in problem.images[i]:
if x[i,s,c].level()[0] > 0.5:
schedule[i].append(c)
break
saver(schedule)
def __residual(matrix, rhs, expr):
"""
Introduce the residual matrix*expr - rhs
"""
return Expr.sub(__mat_vec_prod(matrix, expr), rhs)
z_2_a = m.variable('z_2_a', *binary)
z_2_b = m.variable('z_2_b', *binary)
z_2_c = m.variable('z_2_c', *binary)
z_2_d = m.variable('z_2_d', *binary)
z_3_b = m.variable('z_3_b', *binary)
z_3_c = m.variable('z_3_c', *binary)
z_3_d = m.variable('z_3_d', *binary)
# Inclusion of an image and a command means that image must
# use all command invocation from the clique.
# For instance:
# (1) z_1_a <= w_1
# (2) z_1_a <= y_a
# (3) z_1_a >= w_1 + y_a - 1
m.constraint('c_1_a_1', Expr.sub(z_1_a, w_1), Domain.lessThan(0.0))
m.constraint('c_1_a_2', Expr.sub(z_1_a, y_a), Domain.lessThan(0.0))
m.constraint('c_1_a_3', Expr.sub(z_1_a, Expr.add([w_1, y_a])), Domain.greaterThan(-1.0))
m.constraint('c_1_b_1', Expr.sub(z_1_b, w_1), Domain.lessThan(0.0))
m.constraint('c_1_b_2', Expr.sub(z_1_b, y_b), Domain.lessThan(0.0))
m.constraint('c_1_b_3', Expr.sub(z_1_b, Expr.add([w_1, y_b])), Domain.greaterThan(-1.0))
m.constraint('c_1_c', Expr.sub(0.0, Expr.add([w_1, y_c])), Domain.greaterThan(-1.0))
m.constraint('c_1_d', Expr.sub(0.0, Expr.add([w_1, y_d])), Domain.greaterThan(-1.0))
m.constraint('c_2_a_1', Expr.sub(z_2_a, w_2), Domain.lessThan(0.0))
m.constraint('c_2_a_2', Expr.sub(z_2_a, y_a), Domain.lessThan(0.0))
m.constraint('c_2_a_3', Expr.sub(z_2_a, Expr.add([w_2, y_a])), Domain.greaterThan(-1.0))
m.constraint('c_2_b_1', Expr.sub(z_2_b, w_2), Domain.lessThan(0.0))
def solve(self, problem, saver):
# Construct model.
self.problem = problem
self.model = model = Model()
if self.time is not None:
model.setSolverParam('mioMaxTime', 60.0 * int(self.time))
# x[1,c] = 1 if the master schedule has (null, c) in its first stage
# x[s,c1,c2] = 1 if the master schedule has (c1, c2) in stage s > 1
x = {}
for s in problem.all_stages:
if s == 1:
# First arc in the individual image path.
for c in problem.commands:
x[1, c] = model.variable('x[1,%s]' % c, 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger())
else:
# Other arcs.
for c1, c2 in product(problem.commands, problem.commands):
if c1 == c2:
continue
x[s, c1, c2] = model.variable('x[%s,%s,%s]' % (s, c1, c2),
1, Domain.inRange(0.0, 1.0),
Domain.isInteger())
smax = max(problem.all_stages)
obj = [0.0]
# TODO: deal with images that do not have the same number of commands.
# t[s,c] is the total time incurred at command c in stage s
t = {}
for s in problem.all_stages:
for c in problem.commands:
t[s, c] = model.variable('t[%s,%s]' % (c, s), 1,
Domain.greaterThan(0.0))
if s == 1:
model.constraint(
't[1,%s]' % c,
Expr.sub(t[1, c],
Expr.mul(float(problem.commands[c]), x[1,
c])),
Domain.greaterThan(0.0))
else:
rhs = [0.0]
for c1, coeff in problem.commands.items():
if c1 == c:
continue
else:
rhs = Expr.add(
rhs, Expr.mulElm(t[s - 1, c1], x[s, c1, c]))
model.constraint('t[%s,%s]' % (s, c),
Expr.sub(t[1, c], rhs),
Domain.greaterThan(0.0))
# Objective function = sum of aggregate comand times
if s == smax:
obj = Expr.add(obj, t[s, c])
# y[i,1,c] = 1 if image i starts by going to c
# y[i,s,c1,c2] = 1 if image i goes from command c1 to c2 in stage s > 1
y = {}
for i, cmds in problem.images.items():
for s in problem.stages[i]:
if s == 1:
# First arc in the individual image path.
for c in cmds:
y[i, 1, c] = model.variable('y[%s,1,%s]' % (i, c), 1,
Domain.inRange(0.0, 1.0),
Domain.isInteger())
model.constraint('x_y[i%s,1,c%s]' % (i, c),
Expr.sub(x[1, c], y[i, 1, c]),
Domain.greaterThan(0.0))
else:
# Other arcs.
for c1, c2 in product(cmds, cmds):
if c1 == c2:
continue
y[i, s, c1, c2] = model.variable(
'y[%s,%s,%s,%s]' % (i, s, c1, c2), 1,
Domain.inRange(0.0, 1.0), Domain.isInteger())
model.constraint(
'x_y[i%s,s%s,c%s,c%s]' % (i, s, c1, c2),
Expr.sub(x[s, c1, c2], y[i, s, c1, c2]),
Domain.greaterThan(0.0))
for c in cmds:
# Each command is an arc destination exactly once.
arcs = [y[i, 1, c]]
for c1 in cmds:
if c1 == c:
continue
arcs.extend(
[y[i, s, c1, c] for s in problem.stages[i][1:]])
model.constraint('y[i%s,c%s]' % (i, c), Expr.add(arcs),
Domain.equalsTo(1.0))
# Network balance equations (stages 2 to |stages|-1).
# Sum of arcs in = sum of arcs out.
for s in problem.stages[i][:len(problem.stages[i]) - 1]:
if s == 1:
arcs_in = [y[i, 1, c]]
else:
arcs_in = [y[i, s, c1, c] for c1 in cmds if c1 != c]
arcs_out = [y[i, s + 1, c, c2] for c2 in cmds if c2 != c]
model.constraint(
'y[i%s,s%s,c%s]' % (i, s, c),
Expr.sub(Expr.add(arcs_in), Expr.add(arcs_out)),
Domain.equalsTo(0.0))
model.objective('z', ObjectiveSense.Minimize, Expr.add(x.values()))
# model.objective('z', ObjectiveSense.Minimize, obj)
model.setLogHandler(sys.stdout)
model.acceptedSolutionStatus(AccSolutionStatus.Feasible)
model.solve()
# Create optimal schedule.
schedule = defaultdict(list)
for i, cmds in problem.images.items():
for s in problem.stages[i]:
if s == 1:
# First stage starts our walk.
for c in cmds:
if y[i, s, c].level()[0] > 0.5:
schedule[i].append(c)
break
else:
# After that we know what our starting point is.
for c2 in cmds:
if c2 == c:
continue
if y[i, s, c, c2].level()[0] > 0.5:
schedule[i].append(c2)
c = c2
break
saver(schedule)
def __residual(matrix, rhs, expr):
"""
Introduce the residual matrix*expr - rhs
"""
return Expr.sub(mMath.mat_vec_prod(matrix, expr), rhs)
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
