def solver(fobj, x0, lb=None, ub=None, options={}, method="lmmcp", jac="default", verbose=False):
in_shape = x0.shape
ffobj = lambda x: fobj(x.reshape(in_shape)).flatten()
if not isinstance(jac, str):
pp = np.prod(in_shape)
def Dffobj(t):
tt = t.reshape(in_shape)
dval = jac(tt)
return dval.reshape((pp, pp))
elif jac == "precise":
from numdifftools import Jacobian
Dffobj = Jacobian(ffobj)
else:
Dffobj = MyJacobian(ffobj)
if method == "fsolve":
import scipy.optimize as optimize
factor = options.get("factor")
factor = factor if factor else 1
[sol, infodict, ier, msg] = optimize.fsolve(
ffobj, x0.flatten(), fprime=Dffobj, factor=factor, full_output=True, xtol=1e-10, epsfcn=1e-9
)
if ier != 1:
print msg
elif method == "anderson":
import scipy.optimize as optimize
sol = optimize.anderson(ffobj, x0.flatten())
elif method == "newton_krylov":
import scipy.optimize as optimize
sol = optimize.newton_krylov(ffobj, x0.flatten())
elif method == "lmmcp":
from dolo.numeric.extern.lmmcp import lmmcp, Big
if lb == None:
lb = -Big * np.ones(len(x0.flatten()))
else:
lb = lb.flatten()
if ub == None:
ub = Big * np.ones(len(x0.flatten()))
else:
ub = ub.flatten()
sol = lmmcp(ffobj, Dffobj, x0.flatten(), lb, ub, verbose=verbose, options=options)
return sol.reshape(in_shape)
def test_josephy(self):
from dolo.numeric.extern import lmmcp
x0=np.array( [1.25, 0.00, 0.00, 0.50] )
lb=np.array( [0.00, 0.00, 0.00, 0.00] )
ub=np.array( [1e20, 1e20, 1e20, 1e20] )
resp = lmmcp.lmmcp(josephy, Djosephy, x0, lb, ub)
sol = np.array([1.224746243, -0.0000, 0.0000, 0.5000])
assert( abs(sol - resp).max()<1e-6 )
def solve_lcp(q, J, lb, x0, options={}, verbose=True):
def fun(x):
res = q + J @ x
return res
#
def dfun(x):
return J
ub = numpy.zeros_like(lb) + numpy.inf
opts = {'preprocess': True, 'eps2': 1e-10, 'presteps': 40}
opts.update(options)
from dolo.numeric.extern.lmmcp import lmmcp
res = lmmcp(fun, dfun, x0, lb, ub, verbose=verbose, options=opts)
return res
def solve(self,
verbose=False,
jacs=None,
linear=True,
init_guess={},
return_problem=False,
solver_options={},
xb0=0.1):
if linear:
model = self
import numpy
if jacs is None:
constants, mat, lb = self.compute_jacobian()
else:
constants, mat, lb = jacs
J = -numpy.array(mat).astype(dtype=float)
q = numpy.array(constants).astype(dtype=float).flatten()
lb = numpy.array(lb).astype(dtype=float).flatten()
lb[lb < -1000] = -numpy.inf
x0 = q * 0 + xb0
ub = numpy.zeros_like(lb) + numpy.inf
if return_problem:
return {"q": q, "J": J, "x0": x0, "lb": lb, "ub": ub}
res = solve_lcp(q,
J,
lb,
x0,
options=solver_options,
verbose=verbose)
from collections import OrderedDict
return OrderedDict(zip(map(str, model.full_variables), res))
else:
model = self
import numpy
constants, mat, lb = self.compute_jacobian(
values=init_guess) # warmup
lb[lb < -1000] = -numpy.inf
v = numpy.array(constants).astype(dtype=float).flatten()
if len(init_guess) > 0:
x0 = numpy.array([init_guess[s] for s in self.full_variables])
else:
x0 = v * 0 + 0.1
ub = numpy.zeros_like(lb) + numpy.inf
def resid(x):
vals = {self.full_variables[i]: x[i] for i in range(len(x))}
constants, mat, trash = self.compute_jacobian(values=vals)
mm = numpy.array(mat).astype(dtype=float)
v = numpy.array(constants).astype(dtype=float).flatten()
# lb = numpy.array(lb).astype(dtype=float).flatten()
# res = v-numpy.dot(mm, x)
# return res
return [v, -mm]
def fun(x):
return resid(x)[0]
#
def dfun(x):
return resid(x)[1]
from dolo.numeric.extern.lmmcp import lmmcp
res = lmmcp(fun,
dfun,
x0,
lb,
ub,
verbose=verbose,
options={
'preprocess': True,
'eps2': 1e-10
})
from collections import OrderedDict
return OrderedDict(zip(map(str, model.full_variables), res))
def deterministic_solve(
model,
exogenous=None,
s0=None,
m0=None,
T=100,
ignore_constraints=False,
maxit=100,
initial_guess=None,
verbose=True,
solver="ncpsolve",
keep_steady_state=False,
s1=None, # deprecated
shocks=None, # deprecated
tol=1e-6,
):
"""
Computes a perfect foresight simulation using a stacked-time algorithm.
Typical simulation exercises are:
- start from an out-of-equilibrium exogenous and/or endogenous state: specify `s0` and or `m0`. Missing values are taken from the calibration (`model.calibration`).
- specify an exogenous path for shocks `exogenous`. Initial exogenous state `m0` is then first value of exogenous values. Economy is supposed to have been at the equilibrium for $t<0$, which pins
down initial endogenous state `s0`. `x0` is a jump variable.
If $s0$ is not specified it is then set
equal to the steady-state consistent with the first value
The initial state is specified either by providing a series of exogenous
shocks and assuming the model is initially in equilibrium with the first
value of the shock, or by specifying an initial value for the states.
Parameters
----------
model : Model
Model to be solved
exogenous : array-like, dict, or pandas.DataFrame
A specification for the path of exogenous variables (aka shocks). Can be any of the
following (note by "declaration order" below we mean the order
of `model.symbols["exogenous"]`):
- A 1d numpy array-like specifying a time series for a single
exogenous variable, or all exogenous variables stacked into a single array.
- A 2d numpy array where each column specifies the time series
for one of the shocks in declaration order. This must be an
`N` by number of shocks 2d array.
- A dict where keys are strings found in
`model.symbols["exogenous"]` and values are a time series of
values for that shock. For exogenous variables that do not appear in
this dict, the shock is set to the calibrated value. Note
that this interface is the most flexible as it allows the user
to pass values for only a subset of the model shocks and it
allows the passed time series to be of different lengths.
- A DataFrame where columns map shock names into time series.
The same assumptions and behavior that are used in the dict
case apply here
If nothing is given here, `exogenous` is set equal to the
calibrated values found in `model.calibration["exogenous"]` for
all periods.
If the length of any time-series in shocks is less than `T`
(see below) it is assumed that that particular shock will
remain at the final given value for the duration of the
simulation.
s0 : None or ndarray or dict
If vector with the value of initial states
If an exogenous timeseries is given for exogenous shocks, `s0` will be computed as the steady-state value that is consistent with its first value.
T : int
horizon for the perfect foresight simulation
maxit : int
maximum number of iteration for the nonlinear solver
verbose : boolean
if True, the solver displays iterations
tol : float
stopping criterium for the nonlinear solver
ignore_constraints : bool
if True, complementarity constraints are ignored.
keep_steady_state : bool
if True, initial steady-states and steady-controls are appended to the simulation with date -1.
Returns
-------
pandas dataframe
a dataframe with T+1 observations of the model variables along the
simulation (states, controls, auxiliaries). The simulation should return to a steady-state
consistent with the last specified value of the exogenous shocks.
"""
if shocks is not None:
import warnings
warnings.warn(
"`shocks` argument is deprecated. Use `exogenous` instead.")
exogenous = shocks
if s1 is not None:
import warnings
warnings.warn("`s1` argument is deprecated. Use `s0` instead.")
s0 = s1
# definitions
n_s = len(model.calibration["states"])
n_x = len(model.calibration["controls"])
p = model.calibration["parameters"]
if exogenous is not None:
epsilons = _shocks_to_epsilons(model, exogenous, T)
m0 = epsilons[0, :]
# get initial steady-state
from dolo.algos.steady_state import find_steady_state
start_state = find_steady_state(model, m=m0)
s0 = start_state["states"]
x0 = start_state["controls"]
m1 = epsilons[1, :]
s1 = model.functions["transition"](m0, s0, x0, m1, p)
else:
if s0 is None:
s0 = model.calibration["states"]
if m0 is None:
m0 = model.calibration["exogenous"]
# if m0 is None:
# m0 = np.zeros(len(model.symbols['exogenous']))
# we should probably do something here with the nature of the exogenous process if specified
# i.e. compute nonlinear irf
epsilons = _shocks_to_epsilons(model, exogenous, T)
x0 = model.calibration["controls"]
m1 = epsilons[1, :]
s1 = model.functions["transition"](m0, s0, x0, m1, p)
s1 = np.array(s1)
x1_g = model.calibration["controls"] # we can do better here
sT_g = model.calibration["states"] # we can do better here
xT_g = model.calibration["controls"] # we can do better here
if initial_guess is None:
start = np.concatenate([s1, x1_g])
final = np.concatenate([sT_g, xT_g])
initial_guess = np.row_stack(
[start * (1 - l) + final * l for l in linspace(0.0, 1.0, T + 1)])
else:
if isinstance(initial_guess, pd.DataFrame):
initial_guess = np.array(initial_guess[model.symbols["states"] +
model.symbols["controls"]])
initial_guess = initial_guess[1:, :]
initial_guess = initial_guess[:, :n_s + n_x]
sh = initial_guess.shape
if model.x_bounds and not ignore_constraints:
initial_states = initial_guess[:, :n_s]
[lb,
ub] = [u(epsilons[:, :], initial_states, p) for u in model.x_bounds]
lower_bound = initial_guess * 0 - np.inf
lower_bound[:, n_s:] = lb
upper_bound = initial_guess * 0 + np.inf
upper_bound[:, n_s:] = ub
test1 = max(lb.max(axis=0) - lb.min(axis=0))
test2 = max(ub.max(axis=0) - ub.min(axis=0))
if test1 > 0.00000001 or test2 > 0.00000001:
msg = "Not implemented: perfect foresight solution requires that "
msg += "controls have constant bounds."
raise Exception(msg)
else:
ignore_constraints = True
lower_bound = None
upper_bound = None
print(initial_guess.shape)
print(epsilons.shape)
det_residual(model, initial_guess, s1, xT_g, epsilons)
if not ignore_constraints:
def ff(vec):
return det_residual(model,
vec.reshape(sh),
s1,
xT_g,
epsilons,
jactype="sparse")
v0 = initial_guess.ravel()
if solver == "ncpsolve":
sol, nit = ncpsolve(
ff,
lower_bound.ravel(),
upper_bound.ravel(),
initial_guess.ravel(),
verbose=verbose,
maxit=maxit,
tol=tol,
jactype="sparse",
)
else:
from dolo.numeric.extern.lmmcp import lmmcp
sol = lmmcp(
lambda u: ff(u)[0],
lambda u: ff(u)[1].todense(),
lower_bound.ravel(),
upper_bound.ravel(),
initial_guess.ravel(),
verbose=verbose,
)
nit = -1
sol = sol.reshape(sh)
else:
def ff(vec):
ll = det_residual(model,
vec.reshape(sh),
s1,
xT_g,
epsilons,
diff=True)
return ll
v0 = initial_guess.ravel()
# from scipy.optimize import root
# sol = root(ff, v0, jac=True)
# sol = sol.x.reshape(sh)
from dolo.numeric.optimize.newton import newton
sol, nit = newton(ff, v0, jactype="sparse")
sol = sol.reshape(sh)
# sol = sol[:-1, :]
if (exogenous is not None) and keep_steady_state:
sx = np.concatenate([s0, x0])
sol = np.concatenate([sx[None, :], sol], axis=0)
epsilons = np.concatenate([epsilons[:1:], epsilons], axis=0)
index = range(-1, T + 1)
else:
index = range(0, T + 1)
# epsilons = np.concatenate([epsilons[:1,:], epsilons], axis=0)
if "auxiliary" in model.functions:
colnames = (model.symbols["states"] + model.symbols["controls"] +
model.symbols["auxiliaries"])
# compute auxiliaries
y = model.functions["auxiliary"](epsilons, sol[:, :n_s], sol[:, n_s:],
p)
sol = np.column_stack([sol, y])
else:
colnames = model.symbols["states"] + model.symbols["controls"]
sol = np.column_stack([sol, epsilons])
colnames = colnames + model.symbols["exogenous"]
ts = pd.DataFrame(sol, columns=colnames, index=index)
return ts
def deterministic_solve(model,
shocks=None,
s1=None,
T=100,
ignore_constraints=False,
maxit=100,
initial_guess=None,
verbose=True,
solver='ncpsolve',
tol=1e-6):
"""
Computes a perfect foresight simulation using a stacked-time algorithm.
The initial state is specified either by providing a series of exogenous
shocks and assuming the model is initially in equilibrium with the first
value of the shock, or by specifying an initial value for the states.
Parameters
----------
model : Model
Model to be solved
shocks : array-like, dict, or pandas.DataFrame
A specification of the shocks to the model. Can be any of the
following (note by "declaration order" below we mean the order
of `model.symbols["shocks"]`):
- A 1d numpy array-like specifying a time series for a single
shock, or all shocks stacked into a single array.
- A 2d numpy array where each column specifies the time series
for one of the shocks in declaration order. This must be an
`N` by number of shocks 2d array.
- A dict where keys are strings found in
`model.symbols["shocks"]` and values are a time series of
values for that shock. For model shocks that do not appear in
this dict, the shock is set to the calibrated value. Note
that this interface is the most flexible as it allows the user
to pass values for only a subset of the model shocks and it
allows the passed time series to be of different lengths.
- A DataFrame where columns map shock names into time series.
The same assumptions and behavior that are used in the dict
case apply here
If nothing is given here, `shocks` is set equal to the
calibrated values found in `model.calibration["shocks"]` for
all periods.
If the length of any time-series in shocks is less than `T`
(see below) it is assumed that that particular shock will
remain at the final given value for the duration of the
simulaiton.
s1 : ndarray or dict
a vector with the value of initial states
T : int
horizon for the perfect foresight simulation
maxit : int
maximum number of iteration for the nonlinear solver
verbose : boolean
if True, the solver displays iterations
tol : float
stopping criterium for the nonlinear solver
ignore_constraints : bool
if True, complementarity constraints are ignored.
Returns
-------
pandas dataframe
a dataframe with T+1 observations of the model variables along the
simulation (states, controls, auxiliaries). The first observation is
the steady-state corresponding to the first value of the shocks. The
simulation should return to a steady-state corresponding to the last
value of the exogenous shocks.
"""
# definitions
n_s = len(model.calibration['states'])
n_x = len(model.calibration['controls'])
p = model.calibration['parameters']
epsilons = _shocks_to_epsilons(model, shocks, T)
m0 = epsilons[0, :]
# get initial steady-state
from dolo.algos.steady_state import find_steady_state
# TODO: use initial_guess for steady_state
# TODO:
if s1 is None:
start_state = find_steady_state(model, m=m0)
s0 = start_state['states']
x0 = start_state['controls']
m1 = epsilons[1, :]
s1 = model.functions['transition'](m0, s0, x0, m1, p)
else:
s0 = model.calibration['states'] * np.nan
x0 = model.calibration['controls'] * np.nan
s1 = np.array(s1)
x1_g = model.calibration['controls'] # we can do better here
sT_g = model.calibration['states'] # we can do better here
xT_g = model.calibration['controls'] # we can do better here
if initial_guess is None:
start = np.concatenate([s1, x1_g])
final = np.concatenate([sT_g, xT_g])
initial_guess = np.row_stack(
[start * (1 - l) + final * l for l in linspace(0.0, 1.0, T)])
else:
if isinstance(initial_guess, pd.DataFrame):
initial_guess = np.array(initial_guess[model.symbols['states'] +
model.symbols['controls']])
initial_guess = initial_guess[1:, :]
initial_guess = initial_guess[:, :n_s + n_x]
sh = initial_guess.shape
if model.x_bounds and not ignore_constraints:
initial_states = initial_guess[:, :n_s]
[lb,
ub] = [u(epsilons[1:, :], initial_states, p) for u in model.x_bounds]
lower_bound = initial_guess * 0 - np.inf
lower_bound[:, n_s:] = lb
upper_bound = initial_guess * 0 + np.inf
upper_bound[:, n_s:] = ub
test1 = max(lb.max(axis=0) - lb.min(axis=0))
test2 = max(ub.max(axis=0) - ub.min(axis=0))
if test1 > 0.00000001 or test2 > 0.00000001:
msg = "Not implemented: perfect foresight solution requires that "
msg += "controls have constant bounds."
raise Exception(msg)
else:
ignore_constraints = True
lower_bound = None
upper_bound = None
if not ignore_constraints:
def ff(vec):
return det_residual(model,
vec.reshape(sh),
s1,
xT_g,
epsilons[1:, :],
jactype='sparse')
v0 = initial_guess.ravel()
if solver == 'ncpsolve':
sol, nit = ncpsolve(ff,
lower_bound.ravel(),
upper_bound.ravel(),
initial_guess.ravel(),
verbose=verbose,
maxit=maxit,
tol=tol,
jactype='sparse')
else:
from dolo.numeric.extern.lmmcp import lmmcp
sol = lmmcp(lambda u: ff(u)[0],
lambda u: ff(u)[1].todense(),
lower_bound.ravel(),
upper_bound.ravel(),
initial_guess.ravel(),
verbose=verbose)
nit = -1
sol = sol.reshape(sh)
else:
def ff(vec):
ll = det_residual(model,
vec.reshape(sh),
s1,
xT_g,
epsilons[1:, :],
diff=True)
return (ll)
v0 = initial_guess.ravel()
# from scipy.optimize import root
# sol = root(ff, v0, jac=True)
# sol = sol.x.reshape(sh)
from dolo.numeric.optimize.newton import newton
sol, nit = newton(ff, v0, jactype='sparse')
sol = sol.reshape(sh)
sx = np.concatenate([s0, x0])
# sol = sol[:-1, :]
sol = np.concatenate([sx[None, :], sol], axis=0)
# epsilons = np.concatenate([epsilons[:1,:], epsilons], axis=0)
if 'auxiliary' in model.functions:
colnames = (model.symbols['states'] + model.symbols['controls'] +
model.symbols['auxiliaries'])
# compute auxiliaries
y = model.functions['auxiliary'](epsilons, sol[:, :n_s], sol[:, n_s:],
p)
sol = np.column_stack([sol, y])
else:
colnames = model.symbols['states'] + model.symbols['controls']
sol = np.column_stack([sol, epsilons])
colnames = colnames + model.symbols['exogenous']
ts = pd.DataFrame(sol, columns=colnames)
return ts
def solver(fobj,
x0,
lb=None,
ub=None,
options={},
method='lmmcp',
jac='default',
verbose=False):
in_shape = x0.shape
ffobj = lambda x: fobj(x.reshape(in_shape)).flatten()
if not isinstance(jac, str):
pp = np.prod(in_shape)
def Dffobj(t):
tt = t.reshape(in_shape)
dval = jac(tt)
return dval.reshape((pp, pp))
elif jac == 'precise':
from numdifftools import Jacobian
Dffobj = Jacobian(ffobj)
else:
Dffobj = MyJacobian(ffobj)
if method == 'fsolve':
import scipy.optimize as optimize
factor = options.get('factor')
factor = factor if factor else 1
[sol, infodict, ier, msg] = optimize.fsolve(ffobj,
x0.flatten(),
fprime=Dffobj,
factor=factor,
full_output=True,
xtol=1e-10,
epsfcn=1e-9)
if ier != 1:
print msg
elif method == 'anderson':
import scipy.optimize as optimize
sol = optimize.anderson(ffobj, x0.flatten())
elif method == 'newton_krylov':
import scipy.optimize as optimize
sol = optimize.newton_krylov(ffobj, x0.flatten())
elif method == 'lmmcp':
from dolo.numeric.extern.lmmcp import lmmcp, Big
if lb == None:
lb = -Big * np.ones(len(x0.flatten()))
else:
lb = lb.flatten()
if ub == None:
ub = Big * np.ones(len(x0.flatten()))
else:
ub = ub.flatten()
sol = lmmcp(ffobj,
Dffobj,
x0.flatten(),
lb,
ub,
verbose=verbose,
options=options)
return sol.reshape(in_shape)
def solver(fobj,
x0,
lb=None,
ub=None,
jac=None,
method='lmmcp',
infos=False,
serial_problem=False,
verbose=False,
options={}):
in_shape = x0.shape
if serial_problem:
ffobj = fobj
else:
ffobj = lambda x: fobj(x.reshape(in_shape)).flatten()
# standardize jacobian
if jac is not None:
if not serial_problem:
pp = np.prod(in_shape)
def Dffobj(t):
tt = t.reshape(in_shape)
dval = jac(tt)
return dval.reshape((pp, pp))
else:
Dffobj = jac
elif serial_problem:
from dolo.numeric.newton import SerialDifferentiableFunction
Dffobj = SerialDifferentiableFunction(fobj, in_shape)
else:
Dffobj = MyJacobian(ffobj)
if lb == None:
lb = -np.inf * np.ones(len(x0.flatten()))
if ub == None:
ub = np.inf * np.ones(len(x0.flatten())).flatten()
if not serial_problem:
lb = lb.flatten()
ub = ub.flatten()
if not serial_problem:
x0 = x0.flatten()
if method == 'fsolve':
import scipy.optimize as optimize
factor = options.get('factor')
factor = factor if factor else 1
[sol, infodict, ier, msg] = optimize.fsolve(ffobj,
x0,
fprime=Dffobj,
factor=factor,
full_output=True,
xtol=1e-10,
epsfcn=1e-9)
if ier != 1:
print(msg)
elif method == 'newton':
from dolo.numeric.newton import serial_newton as newton_solver
fun = lambda x: [ffobj(x), Dffobj(x)]
[sol, nit] = newton_solver(fun, x0, verbose=verbose)
elif method == 'lmmcp':
from dolo.numeric.extern.lmmcp import lmmcp
sol = lmmcp(lambda t: -ffobj(t),
lambda u: -Dffobj(u),
x0,
lb,
ub,
verbose=verbose,
options=options)
elif method == 'ncpsolve':
from dolo.numeric.ncpsolve import ncpsolve
fun = lambda x: [ffobj(x), Dffobj(x)]
if serial_problem:
jactype = 'serial'
[sol, nit] = ncpsolve(fun,
lb,
ub,
x0,
verbose=verbose,
infos=True,
jactype='serial')
else:
raise Exception('Unknown method : ' + str(method))
sol = sol.reshape(in_shape)
if infos:
return [sol, nit]
else:
return sol
def solver(fobj, x0, lb=None, ub=None, jac=None, method='lmmcp', infos=False, serial_problem=False, verbose=False, options={}):
in_shape = x0.shape
if serial_problem:
ffobj = fobj
else:
ffobj = lambda x: fobj(x.reshape(in_shape)).flatten()
# standardize jacobian
if jac is not None:
if not serial_problem:
pp = np.prod(in_shape)
def Dffobj(t):
tt = t.reshape(in_shape)
dval = jac(tt)
return dval.reshape( (pp,pp) )
else:
Dffobj = jac
elif serial_problem:
Dffobj = MySerialJacobian(fobj, in_shape)
else:
Dffobj = MyJacobian(ffobj)
if lb == None:
lb = -np.inf*np.ones(len(x0.flatten()))
if ub == None:
ub = np.inf*np.ones(len(x0.flatten())).flatten()
if not serial_problem:
lb = lb.flatten()
ub = ub.flatten()
if not serial_problem:
x0 = x0.flatten()
if method == 'fsolve':
import scipy.optimize as optimize
factor = options.get('factor')
factor = factor if factor else 1
[sol,infodict,ier,msg] = optimize.fsolve(ffobj, x0, fprime=Dffobj, factor=factor, full_output=True, xtol=1e-10, epsfcn=1e-9)
if ier != 1:
print(msg)
elif method == 'newton':
from dolo.numeric.newton import newton_solver
fun = lambda x: [ffobj(x), Dffobj(x) ]
[sol,nit] = newton_solver(fun,x0, verbose=verbose, infos=True)
elif method == 'lmmcp':
from dolo.numeric.extern.lmmcp import lmmcp
sol = lmmcp(lambda t: -ffobj(t), lambda u: -Dffobj(u),x0,lb,ub,verbose=verbose,options=options)
elif method == 'ncpsolve':
from dolo.numeric.ncpsolve import ncpsolve
fun = lambda x: [ffobj(x), Dffobj(x) ]
[sol,nit] = ncpsolve(fun,lb,ub,x0, verbose=verbose, infos=True, serial=serial_problem)
sol = sol.reshape(in_shape)
if infos:
return [sol, nit]
else:
return sol