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_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 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 _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 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 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_riskobjective(exp_ret, covariance_mat, bound):
# define model
model = mModel.build_model("mean_var")
# set of n weights (unconstrained)
weights = mModel.weights(model, "weights", n=len(exp_ret))
# standard deviation induced by covariance matrix
stdev = mMath.stdev(model, "std", weights, covariance_mat)
# impose a bound on this standard deviation
mBound.upper(model, stdev, bound)
mModel.maximise(model=model, expr=Expr.dot(exp_ret, weights))
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 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 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 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))
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)
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 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
