def minimum_variance(matrix):
# Given the matrix of returns a (each column is a series of returns) this method
# computes the weights for a minimum variance portfolio, e.g.
# min 2-Norm[a*w]^2
# s.t.
# w >= 0
# sum[w] = 1
# This is the left-most point on the efficiency frontier in the classic Markowitz theory
# build the model
with Model("Minimum Variance") as model:
# introduce the weight variable
weights = model.variable("weights", matrix.shape[1],
Domain.inRange(0.0, 1.0))
# sum of weights has to be 1
model.constraint(Expr.sum(weights), Domain.equalsTo(1.0))
# returns
r = Expr.mul(Matrix.dense(matrix), weights)
# compute l2_norm squared of those returns
# minimize this l2_norm
model.objective(ObjectiveSense.Minimize,
__l2_norm_squared(model, "2-norm^2(r)", expr=r))
# solve the problem
model.solve()
# return the series of weights
return np.array(weights.level())
def lsq_pos(matrix, rhs):
"""
min 2-norm (matrix*w - rhs)^2
s.t. e'w = 1
w >= 0
"""
# define model
with Model('lsqPos') 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))
v = __l2_norm(model,
"2-norm(res)",
expr=__residual(matrix, rhs, weights))
# minimization of the residual
model.objective(ObjectiveSense.Minimize, v)
# 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
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 _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 minimum_variance(matrix):
# Given the matrix of returns a (each column is a series of returns) this method
# computes the weights for a minimum variance portfolio, e.g.
# min 2-Norm[a*w]^2
# s.t.
# w >= 0
# sum[w] = 1
# This is the left-most point on the efficiency frontier in the classic Markowitz theory
# build the model
with Model("Minimum Variance") as model:
# introduce the weight variable
weights = model.variable("weights", matrix.shape[1], Domain.inRange(0.0, 1.0))
# sum of weights has to be 1
model.constraint(Expr.sum(weights), Domain.equalsTo(1.0))
# returns
r = Expr.mul(Matrix.dense(matrix), weights)
# compute l2_norm squared of those returns
# minimize this l2_norm
model.objective(ObjectiveSense.Minimize, __l2_norm_squared(model, "2-norm^2(r)", expr=r))
# solve the problem
model.solve()
# return the series of weights
return np.array(weights.level())
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 solve(self, problem, saver):
# Construct model.
self.problem = problem
# Do recursive maximal clique detection.
self.clique_data = clique_data = problem.cliques()
self.model = model = Model()
if self.time is not None:
model.setSolverParam('mioMaxTime', 60.0 * int(self.time))
# Each image needs to run all its commands. This keep track of
# what variables run each command for each image.
self.by_img_cmd = by_img_cmd = defaultdict(list)
# Objective is the total cost of all the commands we run.
self._obj = []
# x[i,c] = 1 if image i incurs the cost of command c directly
self.x = x = {}
for img, cmds in problem.images.items():
for cmd in cmds:
name = 'x[%s,%s]' % (img, cmd)
x[name] = v = self.model.variable(
name,
Domain.inRange(0.0, 1.0),
Domain.isInteger()
)
self._obj.append(Expr.mul(float(problem.commands[cmd]), v))
by_img_cmd[img,cmd].append(v)
# cliques[i] = 1 if clique i is used, 0 otherwise
self.cliques = {}
self._inter = 1
self._update(clique_data)
# Each image has to run each of its commands.
for img_cmd, vlist in by_img_cmd.items():
name = 'img-cmd-%s-%s' % img_cmd
self.model.constraint(
name,
Expr.add(vlist),
Domain.equalsTo(1.0)
)
model.objective('z', ObjectiveSense.Minimize, Expr.add(self._obj))
model.setLogHandler(sys.stdout)
model.acceptedSolutionStatus(AccSolutionStatus.Feasible)
model.solve()
# Translate the output of this to a schedule.
schedule = defaultdict(list)
self._translate(schedule, clique_data)
for name, v in x.items():
img, cmd = name.replace('x[','').replace(']','').split(',')
if v.level()[0] > 0.5:
schedule[img].append(cmd)
saver(schedule)
def Update_Z_Constr(self):
self.Remove_Z_Constr()
Z = self.model.getVariable('Z')
if len(self.constr) == 1:
for key, value in self.constr.items(): # only one iteration
self.model.constraint('BB', Z.index(key),
Domain.equalsTo(value))
if len(self.constr) >= 2:
expression = Expr.vstack(
[Z.index(key) for key in self.constr.keys()])
values = [value for key, value in self.constr.items()]
self.model.constraint('BB', expression, Domain.equalsTo(values))
def __Update_Z_Constr(pure_model: Model, constr_z: Dict[int,
int]) -> Model:
Z = pure_model.getVariable('Z')
if len(constr_z) == 1:
for key, value in constr_z.items(): # only one iteration
pure_model.constraint('BB', Z.index(key),
Domain.equalsTo(value))
if len(constr_z) >= 2:
expression = Expr.vstack([Z.index(key) for key in constr_z.keys()])
values = [value for key, value in constr_z.items()]
pure_model.constraint('BB', expression, Domain.equalsTo(values))
return pure_model
def optimize_with_mosek(self, predicted, today):
"""
使用Mosek来优化构建组合。在测试中Mosek比scipy的单纯形法快约18倍，如果可能请尽量使用Mosek。
但是Mosek是一个商业软件，因此你需要一份授权。如果没有授权的话请使用scipy或optlang。
"""
from mosek.fusion import Expr, Model, ObjectiveSense, Domain, SolutionError
index_weight = self.index_weights.loc[today].fillna(0)
index_weight = index_weight / index_weight.sum()
stocks = list(predicted.index)
with Model("portfolio") as M:
x = M.variable("x", len(stocks),
Domain.inRange(0, self.constraint_config['stocks']))
# 权重总和等于一
M.constraint("sum", Expr.sum(x), Domain.equalsTo(1.0))
# 控制风格暴露
for factor_name, limit in self.constraint_config['factors'].items(
):
factor_data = self.factor_data[factor_name].loc[today]
factor_data = factor_data.fillna(factor_data.mean())
index_exposure = (index_weight * factor_data).sum()
stocks_exposure = factor_data.loc[stocks].values
M.constraint(
factor_name, Expr.dot(stocks_exposure.tolist(), x),
Domain.inRange(index_exposure - limit,
index_exposure + limit))
# 控制行业暴露
for industry_name, limit in self.constraint_config[
'industries'].items():
industry_data = self.industry_data[industry_name].loc[
today].fillna(0)
index_exposure = (index_weight * industry_data).sum()
stocks_exposure = industry_data.loc[stocks].values
M.constraint(
industry_name, Expr.dot(stocks_exposure.tolist(), x),
Domain.inRange(index_exposure - limit,
index_exposure + limit))
# 最大化期望收益率
M.objective("MaxRtn", ObjectiveSense.Maximize,
Expr.dot(predicted.tolist(), x))
M.solve()
weights = pd.Series(list(x.level()), index=stocks)
# try:
# weights = pd.Series(list(x.level()), index=stocks)
# except SolutionError:
# raise RuntimeError("Mosek fail to find a feasible solution @ {}".format(str(today)))
return weights[weights > 0]
def solve(self, problem, saver):
# Construct model.
self.problem = problem
# Do recursive maximal clique detection.
self.clique_data = clique_data = problem.cliques()
self.model = model = Model()
if self.time is not None:
model.setSolverParam('mioMaxTime', 60.0 * int(self.time))
# Each image needs to run all its commands. This keep track of
# what variables run each command for each image.
self.by_img_cmd = by_img_cmd = defaultdict(list)
# Objective is the total cost of all the commands we run.
self._obj = []
# x[i,c] = 1 if image i incurs the cost of command c directly
self.x = x = {}
for img, cmds in problem.images.items():
for cmd in cmds:
name = 'x[%s,%s]' % (img, cmd)
x[name] = v = self.model.variable(name,
Domain.inRange(0.0, 1.0),
Domain.isInteger())
self._obj.append(Expr.mul(float(problem.commands[cmd]), v))
by_img_cmd[img, cmd].append(v)
# cliques[i] = 1 if clique i is used, 0 otherwise
self.cliques = {}
self._inter = 1
self._update(clique_data)
# Each image has to run each of its commands.
for img_cmd, vlist in by_img_cmd.items():
name = 'img-cmd-%s-%s' % img_cmd
self.model.constraint(name, Expr.add(vlist), Domain.equalsTo(1.0))
model.objective('z', ObjectiveSense.Minimize, Expr.add(self._obj))
model.setLogHandler(sys.stdout)
model.acceptedSolutionStatus(AccSolutionStatus.Feasible)
model.solve()
# Translate the output of this to a schedule.
schedule = defaultdict(list)
self._translate(schedule, clique_data)
for name, v in x.items():
img, cmd = name.replace('x[', '').replace(']', '').split(',')
if v.level()[0] > 0.5:
schedule[img].append(cmd)
saver(schedule)
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 __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 _solve_cvxopt_mosek(self):
from mosek.fusion import Model, Domain, ObjectiveSense, Expr
n, r = self.A
with Model('L1 lp formulation') as model:
model.variable('x', r, Domain.greaterThan(0.0))
model.objective('l1 norm', ObjectiveSense.Minimize, Expr.dot())
raise NotImplementedError
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 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 __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 markowitz_riskobjective(exp_ret, covariance_mat, bound):
# define model
with Model("mean var") as model:
# set of n weights (unconstrained)
weights = model.variable("weights", len(exp_ret), Domain.inRange(0.0, 1.0))
# standard deviation induced by covariance matrix
stdev = __stdev(model, "std", weights, covariance_mat)
# impose a bound on this standard deviation
#mBound.upper(model, stdev, bound)
model.constraint(stdev, Domain.lessThan(bound))
#mModel.maximise(model=model, expr=Expr.dot(exp_ret, weights))
model.objective(ObjectiveSense.Maximize, Expr.dot(exp_ret, weights))
# solve the problem
model.solve()
return np.array(weights.level())
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 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 markowitz_riskobjective(exp_ret, covariance_mat, bound):
# define model
with Model("mean var") as model:
# set of n weights (unconstrained)
weights = model.variable("weights", len(exp_ret),
Domain.inRange(0.0, 1.0))
# standard deviation induced by covariance matrix
stdev = __stdev(model, "std", weights, covariance_mat)
# impose a bound on this standard deviation
#mBound.upper(model, stdev, bound)
model.constraint(stdev, Domain.lessThan(bound))
#mModel.maximise(model=model, expr=Expr.dot(exp_ret, weights))
model.objective(ObjectiveSense.Maximize, Expr.dot(exp_ret, weights))
# solve the problem
model.solve()
return np.array(weights.level())
def lsq_ls(matrix, rhs):
"""
min 2-norm (matrix*w - rhs)^2
s.t. e'w = 1
"""
# define model
with Model('lsqPos') as model:
weights = model.variable("weights", matrix.shape[1], Domain.inRange(-np.infty, +np.infty))
# e'*w = 1
model.constraint(Expr.sum(weights), Domain.equalsTo(1.0))
v = __l2_norm(model, "2-norm(res)", expr=__residual(matrix, rhs, weights))
# minimization of the residual
model.objective(ObjectiveSense.Minimize, v)
# solve the problem
model.solve()
return np.array(weights.level())
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_1') as M:
A = Matrix.dense(A)
# variable
X = M.variable('X', Domain.inPSDCone(n))
# objective function
M.objective(ObjectiveSense.Maximize,
Expr.sum(Expr.dot(Matrix.ones(n, n), X)))
# constraints
M.constraint(f'c1', Expr.sum(Expr.dot(X, A)), Domain.equalsTo(0.))
M.constraint(f'c2', Expr.sum(Expr.dot(X, Matrix.eye(n))),
Domain.equalsTo(1.))
# solution
M.solve()
sol = X.level()
return sum(sol)
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 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 __init__(self,name,
foods,
nutrients,
daily_allowance,
nutritive_value):
Model.__init__(self,name)
finished = False
try:
self.foods = [ str(f) for f in foods ]
self.nutrients = [ str(n) for n in nutrients ]
self.dailyAllowance = array.array(daily_allowance, float)
self.nutrientValue = DenseMatrix(nutritive_value).transpose()
M = len(self.foods)
N = len(self.nutrients)
if len(self.dailyAllowance) != N:
raise ValueError("Length of daily_allowance does not match "
"the number of nutrients")
if self.nutrientValue.numColumns() != M:
raise ValueError("Number of rows in nutrient_value does not "
"match the number of foods")
if self.nutrientValue.numRows() != N:
raise ValueError("Number of columns in nutrient_value does "
"not match the number of nutrients")
self.__dailyPurchase = self.variable('Daily Purchase',
StringSet(self.foods),
Domain.greaterThan(0.0))
self.__dailyNutrients = \
self.constraint('Nutrient Balance',
StringSet(nutrients),
Expr.mul(self.nutrientValue,self.__dailyPurchase),
Domain.greaterThan(self.dailyAllowance))
self.objective(ObjectiveSense.Minimize, Expr.sum(self.__dailyPurchase))
finished = True
finally:
if not finished:
self.__del__()
def __init__(self,name,
foods,
nutrients,
daily_allowance,
nutritive_value):
Model.__init__(self,name)
finished = False
try:
self.foods = [ str(f) for f in foods ]
self.nutrients = [ str(n) for n in nutrients ]
self.dailyAllowance = numpy.array(daily_allowance, float)
self.nutrientValue = Matrix.dense(nutritive_value).transpose()
M = len(self.foods)
N = len(self.nutrients)
if len(self.dailyAllowance) != N:
raise ValueError("Length of daily_allowance does not match "
"the number of nutrients")
if self.nutrientValue.numColumns() != M:
raise ValueError("Number of rows in nutrient_value does not "
"match the number of foods")
if self.nutrientValue.numRows() != N:
raise ValueError("Number of columns in nutrient_value does "
"not match the number of nutrients")
self.__dailyPurchase = self.variable('Daily Purchase',
StringSet(self.foods),
Domain.greaterThan(0.0))
self.__dailyNutrients = \
self.constraint('Nutrient Balance',
StringSet(nutrients),
Expr.mul(self.nutrientValue,self.__dailyPurchase),
Domain.greaterThan(self.dailyAllowance))
self.objective(ObjectiveSense.Minimize, Expr.sum(self.__dailyPurchase))
finished = True
finally:
if not finished:
self.__del__()
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 __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 __quad_cone(model, expr1, expr2):
model.constraint(Expr.vstack(expr1, expr2), Domain.inQCone())
import json
import numpy as np
from mosek.fusion import Model, Domain, Expr, ObjectiveSense
A = np.array([[-1., 3.], [4., -1.]])
b = np.array([4., 6.])
c = np.array([1., 1.])
with Model('ex2') as M:
# variable y
y = M.variable('y', 2, Domain.greaterThan(0.))
# constraints
M.constraint('c1', Expr.dot(A.T[0, :], y), Domain.greaterThan(c[0]))
M.constraint('c2', Expr.dot(A.T[1, :], y), Domain.greaterThan(c[1]))
# objective function
M.objective('obj', ObjectiveSense.Minimize, Expr.dot(b, y))
# solve
M.solve()
# solution
sol = y.level()
# report to json
with open('ex2_output.json', 'w') as f:
json.dump(
{
'solution': {f'y[{i+1}]': yi
for i, yi in enumerate(sol)},
'cost': np.dot(b, sol),
}, f)
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 __rotated_quad_cone(model, expr1, expr2, expr3):
model.constraint(Expr.vstack(expr1, expr2, expr3), Domain.inRotatedQCone())
#
# Image 3:
# x_3_b = 0
# x_3_c = 0
# x_3_d = 0
#
# Cliques:
# x_23_bcd = 0
# x_123_b = 1
# x_123_b_23_cd = 1
# x_12_a = 0
r = {'A': 5.0, 'B': 10.0, 'C': 7.0, 'D': 12.0}
m = Model()
binary = (Domain.inRange(0.0, 1.0), Domain.isInteger())
# Provide a variable for each image and command. This is 1 if the command
# is not run as part of a clique for the image.
x_1_a = m.variable('x_1_a', *binary)
x_1_b = m.variable('x_1_b', *binary)
x_2_a = m.variable('x_2_a', *binary)
x_2_b = m.variable('x_2_b', *binary)
x_2_c = m.variable('x_2_c', *binary)
x_2_d = m.variable('x_2_d', *binary)
x_3_b = m.variable('x_3_b', *binary)
x_3_c = m.variable('x_3_c', *binary)
x_3_d = m.variable('x_3_d', *binary)
#
# Image 3:
# x_3_b = 0
# x_3_c = 0
# x_3_d = 0
#
# Cliques:
# x_23_bcd = 0
# x_123_b = 1
# x_123_b_23_cd = 1
# x_12_a = 0
r = {'A': 5.0, 'B': 10.0, 'C': 7.0, 'D': 12.0}
m = Model()
binary = (Domain.inRange(0.0, 1.0), Domain.isInteger())
# Provide a variable for each image and command. This is 1 if the command
# is not run as part of a clique for the image.
x_1_a = m.variable('x_1_a', *binary)
x_1_b = m.variable('x_1_b', *binary)
x_2_a = m.variable('x_2_a', *binary)
x_2_b = m.variable('x_2_b', *binary)
x_2_c = m.variable('x_2_c', *binary)
x_2_d = m.variable('x_2_d', *binary)
x_3_b = m.variable('x_3_b', *binary)
x_3_c = m.variable('x_3_c', *binary)
x_3_d = m.variable('x_3_d', *binary)
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)
# x_2_b = 0
# x_2_c = 0
# x_2_d = 0
#
# Image 3:
# x_3_b = 0
# x_3_c = 0
# x_3_d = 0
#
# Cliques:
# x_23_bcd = 1
r = {'A': 5.0, 'B': 10.0, 'C': 7.0, 'D': 12.0}
m = Model()
binary = (Domain.inRange(0.0, 1.0), Domain.isInteger())
# Provide a variable for each image and command. This is 1 if the command
# is not run as part of a clique for the image.
x_1_a = m.variable('x_1_a', *binary)
x_1_b = m.variable('x_1_b', *binary)
x_2_a = m.variable('x_2_a', *binary)
x_2_b = m.variable('x_2_b', *binary)
x_2_c = m.variable('x_2_c', *binary)
x_2_d = m.variable('x_2_d', *binary)
x_3_b = m.variable('x_3_b', *binary)
x_3_c = m.variable('x_3_c', *binary)
x_3_d = m.variable('x_3_d', *binary)
def __quad_cone(model, expr1, expr2):
model.constraint(Expr.vstack(expr1, expr2), Domain.inQCone())
def __rotated_quad_cone(model, expr1, expr2, expr3):
model.constraint(Expr.vstack(expr1, expr2, expr3), Domain.inRotatedQCone())
import json
import numpy as np
from mosek.fusion import Model, Domain, Expr, ObjectiveSense
A = np.array([[-1., 3.], [4., -1.]])
b = np.array([4., 6.])
c = np.array([1., 1.])
with Model('ex1') as M:
# variable x
x = M.variable('x', 2, Domain.greaterThan(0.))
# constraints
M.constraint('c1', Expr.dot(A[0, :], x), Domain.lessThan(b[0]))
M.constraint('c2', Expr.dot(A[1, :], x), Domain.lessThan(b[1]))
# objective function
M.objective('obj', ObjectiveSense.Maximize, Expr.dot(c, x))
# solve
M.solve()
# solution
sol = x.level()
# report to json
with open('ex1_output.json', 'w') as f:
json.dump({
'solution': {f'x[{i+1}]': xi for i, xi in enumerate(sol)},
'cost': np.dot(sol, c),
}, f)
# y_a = 0
# y_b = 1
# y_c = 0
# y_d = 0
#
# Interactions:
# m_2 = 0
# m_3 = 0
# n_b = 1
# n_c = 0
# n_d = 0
r = {'A': 5.0, 'B': 10.0, 'C': 7.0, 'D': 12.0}
m = Model()
binary = (Domain.inRange(0.0, 1.0), Domain.isInteger())
# Variables to determine if we include commands in the clique.
y_a = m.variable('y_a', *binary)
y_b = m.variable('y_b', *binary)
y_c = m.variable('y_c', *binary)
y_d = m.variable('y_d', *binary)
# Variables to determine if we include images in the clique.
w_1 = m.variable('w_1', *binary)
w_2 = m.variable('w_2', *binary)
w_3 = m.variable('w_3', *binary)
# Variables to enforce relationships between y and w decisions.
z_1_a = m.variable('z_1_a', *binary)
z_1_b = m.variable('z_1_b', *binary)
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 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)
#
# Images:
# w_1 = 0
# w_2 = 1
# w_3 = 1
#
# Commands:
# y_a = 0
# y_b = 1
# y_c = 1
# y_d = 1
r = {'A': 5.0, 'B': 10.0, 'C': 7.0, 'D': 12.0}
m = Model()
binary = (Domain.inRange(0.0, 1.0), Domain.isInteger())
# Variables to determine if we include commands in the clique.
y_a = m.variable('y_a', *binary)
y_b = m.variable('y_b', *binary)
y_c = m.variable('y_c', *binary)
y_d = m.variable('y_d', *binary)
# Variables to determine if we include images in the clique.
w_1 = m.variable('w_1', *binary)
w_2 = m.variable('w_2', *binary)
w_3 = m.variable('w_3', *binary)
# Variables to enforce relationships between y and w decisions.
z_1_a = m.variable('z_1_a', *binary)
z_1_b = m.variable('z_1_b', *binary)
#
# Image 2:
# x_2_a = 1
# x_2_b = 1
# x_2_c = 1
# x_2_d = 1
#
# Image 3:
# x_3_b = 1
# x_3_c = 1
# x_3_d = 1
r = {'A': 5.0, 'B': 10.0, 'C': 7.0, 'D': 12.0}
m = Model()
binary = (Domain.inRange(0.0, 1.0), Domain.isInteger())
# Provide a variable for each image and command. This is 1 if the command
# is not run as part of a clique for the image.
x_1_a = m.variable('x_1_a', *binary)
x_1_b = m.variable('x_1_b', *binary)
x_2_a = m.variable('x_2_a', *binary)
x_2_b = m.variable('x_2_b', *binary)
x_2_c = m.variable('x_2_c', *binary)
x_2_d = m.variable('x_2_d', *binary)
x_3_b = m.variable('x_3_b', *binary)
x_3_c = m.variable('x_3_c', *binary)
x_3_d = m.variable('x_3_d', *binary)
# x_2_d = 0
#
# Image 3:
# x_3_b = 0
# x_3_c = 0
# x_3_d = 0
#
# Cliques:
# x_23_bcd = 0
# x_123_b = 1
# x_123_b_23_cd = 1
r = {'A': 5.0, 'B': 10.0, 'C': 7.0, 'D': 12.0}
m = Model()
binary = (Domain.inRange(0.0, 1.0), Domain.isInteger())
# Provide a variable for each image and command. This is 1 if the command
# is not run as part of a clique for the image.
x_1_a = m.variable('x_1_a', *binary)
x_1_b = m.variable('x_1_b', *binary)
x_2_a = m.variable('x_2_a', *binary)
x_2_b = m.variable('x_2_b', *binary)
x_2_c = m.variable('x_2_c', *binary)
x_2_d = m.variable('x_2_d', *binary)
x_3_b = m.variable('x_3_b', *binary)
x_3_c = m.variable('x_3_c', *binary)
x_3_d = m.variable('x_3_d', *binary)