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 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 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 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)
