def value_iteration(model,
grid={},
tol=1e-6,
maxit=500,
maxit_howard=20,
verbose=False):
"""
Solve for the value function and associated Markov decision rule by iterating over
the value function.
Parameters:
-----------
model :
"dtmscc" model. Must contain a 'felicity' function.
grid :
grid options
dr :
decision rule to evaluate
Returns:
--------
mdr : Markov decision rule
The solved decision rule/policy function
mdrv: decision rule
The solved value function
"""
transition = model.functions['transition']
felicity = model.functions['felicity']
controls_lb = model.functions['controls_lb']
controls_ub = model.functions['controls_ub']
parms = model.calibration['parameters']
discount = model.calibration['beta']
x0 = model.calibration['controls']
m0 = model.calibration['exogenous']
s0 = model.calibration['states']
r0 = felicity(m0, s0, x0, parms)
process = model.exogenous
dprocess = process.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
endo_grid = model.get_grid(**grid)
exo_grid = dprocess.grid
mdrv = DecisionRule(exo_grid, endo_grid)
grid = mdrv.endo_grid.nodes()
N = grid.shape[0]
n_x = len(x0)
mdr = constant_policy(model)
controls_0 = np.zeros((n_ms, N, n_x))
for i_ms in range(n_ms):
controls_0[i_ms, :, :] = mdr.eval_is(i_ms, grid)
values_0 = np.zeros((n_ms, N, 1))
# for i_ms in range(n_ms):
# values_0[i_ms, :, :] = mdrv(i_ms, grid)
mdr = DecisionRule(exo_grid, endo_grid)
# mdr.set_values(controls_0)
# THIRD: value function iterations until convergence
it = 0
err_v = 100
err_v_0 = 0
gain_v = 0.0
err_x = 100
err_x_0 = 0
tol_x = 1e-5
tol_v = 1e-7
itprint = IterationsPrinter(
('N', int), ('Error_V', float), ('Gain_V', float), ('Error_x', float),
('Gain_x', float), ('Eval_n', int), ('Time', float),
verbose=verbose)
itprint.print_header('Start value function iterations.')
while (it < maxit) and (err_v > tol or err_x > tol_x):
t_start = time.time()
it += 1
mdr.set_values(controls_0)
if it > 2:
ev = evaluate_policy(model,
mdr,
initial_guess=mdrv,
verbose=False,
details=True)
else:
ev = evaluate_policy(model, mdr, verbose=False, details=True)
mdrv = ev.solution
for i_ms in range(n_ms):
values_0[i_ms, :, :] = mdrv.eval_is(i_ms, grid)
values = values_0.copy()
controls = controls_0.copy()
for i_m in range(n_ms):
m = dprocess.node(i_m)
for n in range(N):
s = grid[n, :]
x = controls[i_m, n, :]
lb = controls_lb(m, s, parms)
ub = controls_ub(m, s, parms)
bnds = [e for e in zip(lb, ub)]
def valfun(xx):
return -choice_value(transition, felicity, i_m, s, xx,
mdrv, dprocess, parms, discount)[0]
res = scipy.optimize.minimize(valfun, x, bounds=bnds)
controls[i_m, n, :] = res.x
values[i_m, n, 0] = -valfun(x)
# compute error, update value and dr
err_x = abs(controls - controls_0).max()
err_v = abs(values - values_0).max()
t_end = time.time()
elapsed = t_end - t_start
values_0 = values
controls_0 = controls
gain_x = err_x / err_x_0
gain_v = err_v / err_v_0
err_x_0 = err_x
err_v_0 = err_v
itprint.print_iteration(N=it,
Error_V=err_v,
Gain_V=gain_v,
Error_x=err_x,
Gain_x=gain_x,
Eval_n=ev.iterations,
Time=elapsed)
itprint.print_finished()
mdr = DecisionRule(exo_grid, endo_grid)
mdr.set_values(controls)
mdrv.set_values(values_0)
return mdr, mdrv
def simulate(
model: Model,
dr: DecisionRule,
*,
process=None,
N=1,
T=40,
s0=None,
i0=None,
m0=None,
driving_process=None,
seed=42,
stochastic=True,
):
"""Simulate a model using the specified decision rule.
Parameters
----------
model: Model
dr: decision rule
process:
s0: ndarray
initial state where all simulations start
driving_process: ndarray
realization of exogenous driving process (drawn randomly if None)
N: int
number of simulations
T: int
horizon for the simulations
seed: int
used to initialize the random number generator. Use it to replicate
exact same results among simulations
discard: boolean (False)
if True, then all simulations containing at least one non finite value
are discarded
Returns
-------
xarray.DataArray:
returns a ``T x N x n_v`` array where ``n_v``
is the number of variables.
"""
if isinstance(dr, AlgoResult):
dr = dr.dr
calib = model.calibration
parms = numpy.array(calib["parameters"])
if s0 is None:
s0 = calib["states"]
n_x = len(model.symbols["controls"])
n_s = len(model.symbols["states"])
s_simul = numpy.zeros((T, N, n_s))
x_simul = numpy.zeros((T, N, n_x))
s_simul[0, :, :] = s0[None, :]
# are we simulating a markov chain or a continuous process ?
if driving_process is not None:
if len(driving_process.shape) == 3:
m_simul = driving_process
sim_type = "continuous"
if m0 is None:
m0 = model.calibration["exogenous"]
x_simul[0, :, :] = dr.eval_ms(m0[None, :], s0[None, :])[0, :]
elif len(driving_process.shape) == 2:
i_simul = driving_process
nodes = dr.exo_grid.nodes
m_simul = nodes[i_simul]
# inds = i_simul.ravel()
# m_simul = np.reshape( np.concatenate( [nodes[i,:][None,:] for i in inds.ravel()], axis=0 ), inds.shape + (-1,) )
sim_type = "discrete"
x_simul[0, :, :] = dr.eval_is(i0, s0[None, :])[0, :]
else:
raise Exception("Incorrect specification of driving values.")
m0 = m_simul[0, :, :]
else:
from dolo.numeric.processes import DiscreteProcess
if process is None:
if hasattr(dr, "dprocess") and hasattr(dr.dprocess, "simulate"):
process = dr.dprocess
else:
process = model.exogenous
# detect type of simulation
if not isinstance(process, DiscreteProcess):
sim_type = "continuous"
else:
sim_type = "discrete"
if sim_type == "discrete":
if i0 is None:
i0 = 0
dp = process
m_simul = dp.simulate(N, T, i0=i0, stochastic=stochastic)
i_simul = find_index(m_simul, dp.values)
m0 = dp.node(i0)
x0 = dr.eval_is(i0, s0[None, :])[0, :]
else:
m_simul = process.simulate(N, T, m0=m0, stochastic=stochastic)
if isinstance(m_simul, xr.DataArray):
m_simul = m_simul.data
sim_type = "continuous"
if m0 is None:
m0 = model.calibration["exogenous"]
x0 = dr.eval_ms(m0[None, :], s0[None, :])[0, :]
x_simul[0, :, :] = x0[None, :]
f = model.functions["arbitrage"]
g = model.functions["transition"]
numpy.random.seed(seed)
mp = m0
for i in range(T):
m = m_simul[i, :, :]
s = s_simul[i, :, :]
if sim_type == "discrete":
i_m = i_simul[i, :]
xx = [
dr.eval_is(i_m[ii], s[ii, :][None, :])[0, :] for ii in range(s.shape[0])
]
x = np.row_stack(xx)
else:
x = dr.eval_ms(m, s)
x_simul[i, :, :] = x
ss = g(mp, s, x, m, parms)
if i < T - 1:
s_simul[i + 1, :, :] = ss
mp = m
if "auxiliary" not in model.functions: # TODO: find a better test than this
l = [s_simul, x_simul]
varnames = model.symbols["states"] + model.symbols["controls"]
else:
aux = model.functions["auxiliary"]
a_simul = aux(
m_simul.reshape((N * T, -1)),
s_simul.reshape((N * T, -1)),
x_simul.reshape((N * T, -1)),
parms,
)
a_simul = a_simul.reshape(T, N, -1)
l = [m_simul, s_simul, x_simul, a_simul]
varnames = (
model.symbols["exogenous"]
+ model.symbols["states"]
+ model.symbols["controls"]
+ model.symbols["auxiliaries"]
)
simul = numpy.concatenate(l, axis=2)
if sim_type == "discrete":
varnames = ["_i_m"] + varnames
simul = np.concatenate([i_simul[:, :, None], simul], axis=2)
data = xr.DataArray(
simul,
dims=["T", "N", "V"],
coords={"T": range(T), "N": range(N), "V": varnames},
)
return data
def value_iteration(model,
grid={},
tol=1e-6,
maxit=500,
maxit_howard=20,
verbose=False,
details=True):
"""
Solve for the value function and associated Markov decision rule by iterating over
the value function.
Parameters:
-----------
model :
"dtmscc" model. Must contain a 'felicity' function.
grid :
grid options
dr :
decision rule to evaluate
Returns:
--------
mdr : Markov decision rule
The solved decision rule/policy function
mdrv: decision rule
The solved value function
"""
transition = model.functions['transition']
felicity = model.functions['felicity']
controls_lb = model.functions['controls_lb']
controls_ub = model.functions['controls_ub']
parms = model.calibration['parameters']
discount = model.calibration['beta']
x0 = model.calibration['controls']
m0 = model.calibration['exogenous']
s0 = model.calibration['states']
r0 = felicity(m0, s0, x0, parms)
process = model.exogenous
dprocess = process.discretize()
n_ms = dprocess.n_nodes() # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
endo_grid = model.get_grid(**grid)
exo_grid = dprocess.grid
mdrv = DecisionRule(exo_grid, endo_grid)
grid = mdrv.endo_grid.nodes()
N = grid.shape[0]
n_x = len(x0)
mdr = constant_policy(model)
controls_0 = np.zeros((n_ms, N, n_x))
for i_ms in range(n_ms):
controls_0[i_ms, :, :] = mdr.eval_is(i_ms, grid)
values_0 = np.zeros((n_ms, N, 1))
# for i_ms in range(n_ms):
# values_0[i_ms, :, :] = mdrv(i_ms, grid)
mdr = DecisionRule(exo_grid, endo_grid)
# mdr.set_values(controls_0)
# THIRD: value function iterations until convergence
it = 0
err_v = 100
err_v_0 = 0
gain_v = 0.0
err_x = 100
err_x_0 = 0
tol_x = 1e-5
tol_v = 1e-7
itprint = IterationsPrinter(
('N', int), ('Error_V', float), ('Gain_V', float), ('Error_x', float),
('Gain_x', float), ('Eval_n', int), ('Time', float),
verbose=verbose)
itprint.print_header('Start value function iterations.')
while (it < maxit) and (err_v > tol or err_x > tol_x):
t_start = time.time()
it += 1
mdr.set_values(controls_0)
if it > 2:
ev = evaluate_policy(
model, mdr, initial_guess=mdrv, verbose=False, details=True)
else:
ev = evaluate_policy(model, mdr, verbose=False, details=True)
mdrv = ev.solution
for i_ms in range(n_ms):
values_0[i_ms, :, :] = mdrv.eval_is(i_ms, grid)
values = values_0.copy()
controls = controls_0.copy()
for i_m in range(n_ms):
m = dprocess.node(i_m)
for n in range(N):
s = grid[n, :]
x = controls[i_m, n, :]
lb = controls_lb(m, s, parms)
ub = controls_ub(m, s, parms)
bnds = [e for e in zip(lb, ub)]
def valfun(xx):
return -choice_value(transition, felicity, i_m, s, xx,
mdrv, dprocess, parms, discount)[0]
res = scipy.optimize.minimize(valfun, x, bounds=bnds)
controls[i_m, n, :] = res.x
values[i_m, n, 0] = -valfun(x)
# compute error, update value and dr
err_x = abs(controls - controls_0).max()
err_v = abs(values - values_0).max()
t_end = time.time()
elapsed = t_end - t_start
values_0 = values
controls_0 = controls
gain_x = err_x / err_x_0
gain_v = err_v / err_v_0
err_x_0 = err_x
err_v_0 = err_v
itprint.print_iteration(
N=it,
Error_V=err_v,
Gain_V=gain_v,
Error_x=err_x,
Gain_x=gain_x,
Eval_n=ev.iterations,
Time=elapsed)
itprint.print_finished()
mdr = DecisionRule(exo_grid, endo_grid)
mdr.set_values(controls)
mdrv.set_values(values_0)
if not details:
return mdr, mdrv
else:
return ValueIterationResult(
mdr, #:AbstractDecisionRule
mdrv, #:AbstractDecisionRule
it, #:Int
dprocess, #:AbstractDiscretizedProcess
err_x<tol_x, #:Bool
tol_x, #:Float64
err_x, #:Float64
err_v<tol_v, #:Bool
tol_v, #:Float64
err_v, #:Float64
None, #log: #:ValueIterationLog
None #trace: #:Union{Nothing,IterationTrace
)
def value_iteration(
model: Model,
*,
verbose: bool = False, #
details: bool = True, #
tol=1e-6,
maxit=500,
maxit_howard=20,
) -> ValueIterationResult:
"""
Solve for the value function and associated Markov decision rule by iterating over
the value function.
Parameters:
-----------
model :
model to be solved
dr :
decision rule to evaluate
Returns:
--------
mdr : Markov decision rule
The solved decision rule/policy function
mdrv: decision rule
The solved value function
"""
transition = model.functions["transition"]
felicity = model.functions["felicity"]
controls_lb = model.functions["controls_lb"]
controls_ub = model.functions["controls_ub"]
parms = model.calibration["parameters"]
discount = model.calibration["beta"]
x0 = model.calibration["controls"]
m0 = model.calibration["exogenous"]
s0 = model.calibration["states"]
r0 = felicity(m0, s0, x0, parms)
process = model.exogenous
grid, dprocess = model.discretize()
endo_grid = grid["endo"]
exo_grid = grid["exo"]
n_ms = dprocess.n_nodes # number of exogenous states
n_mv = dprocess.n_inodes(
0) # this assume number of integration nodes is constant
mdrv = DecisionRule(exo_grid, endo_grid)
s = mdrv.endo_grid.nodes
N = s.shape[0]
n_x = len(x0)
mdr = constant_policy(model)
controls_0 = np.zeros((n_ms, N, n_x))
for i_ms in range(n_ms):
controls_0[i_ms, :, :] = mdr.eval_is(i_ms, s)
values_0 = np.zeros((n_ms, N, 1))
# for i_ms in range(n_ms):
# values_0[i_ms, :, :] = mdrv(i_ms, grid)
mdr = DecisionRule(exo_grid, endo_grid)
# mdr.set_values(controls_0)
# THIRD: value function iterations until convergence
it = 0
err_v = 100
err_v_0 = 0
gain_v = 0.0
err_x = 100
err_x_0 = 0
tol_x = 1e-5
tol_v = 1e-7
itprint = IterationsPrinter(
("N", int),
("Error_V", float),
("Gain_V", float),
("Error_x", float),
("Gain_x", float),
("Eval_n", int),
("Time", float),
verbose=verbose,
)
itprint.print_header("Start value function iterations.")
while (it < maxit) and (err_v > tol or err_x > tol_x):
t_start = time.time()
it += 1
mdr.set_values(controls_0)
if it > 2:
ev = evaluate_policy(model,
mdr,
dr0=mdrv,
verbose=False,
details=True)
else:
ev = evaluate_policy(model, mdr, verbose=False, details=True)
mdrv = ev.solution
for i_ms in range(n_ms):
values_0[i_ms, :, :] = mdrv.eval_is(i_ms, s)
values = values_0.copy()
controls = controls_0.copy()
for i_m in range(n_ms):
m = dprocess.node(i_m)
for n in range(N):
s_ = s[n, :]
x = controls[i_m, n, :]
lb = controls_lb(m, s_, parms)
ub = controls_ub(m, s_, parms)
bnds = [e for e in zip(lb, ub)]
def valfun(xx):
return -choice_value(
transition,
felicity,
i_m,
s_,
xx,
mdrv,
dprocess,
parms,
discount,
)[0]
res = scipy.optimize.minimize(valfun, x, bounds=bnds)
controls[i_m, n, :] = res.x
values[i_m, n, 0] = -valfun(x)
# compute error, update value and dr
err_x = abs(controls - controls_0).max()
err_v = abs(values - values_0).max()
t_end = time.time()
elapsed = t_end - t_start
values_0 = values
controls_0 = controls
gain_x = err_x / err_x_0
gain_v = err_v / err_v_0
err_x_0 = err_x
err_v_0 = err_v
itprint.print_iteration(
N=it,
Error_V=err_v,
Gain_V=gain_v,
Error_x=err_x,
Gain_x=gain_x,
Eval_n=ev.iterations,
Time=elapsed,
)
itprint.print_finished()
mdr = DecisionRule(exo_grid, endo_grid)
mdr.set_values(controls)
mdrv.set_values(values_0)
if not details:
return mdr, mdrv
else:
return ValueIterationResult(
mdr, #:AbstractDecisionRule
mdrv, #:AbstractDecisionRule
it, #:Int
dprocess, #:AbstractDiscretizedProcess
err_x < tol_x, #:Bool
tol_x, #:Float64
err_x, #:Float64
err_v < tol_v, #:Bool
tol_v, #:Float64
err_v, #:Float64
None, # log: #:ValueIterationLog
None, # trace: #:Union{Nothing,IterationTrace
)
def improved_time_iteration(
model: Model,
*,
dr0: DecisionRule = None, #
verbose: bool = True, #
details: bool = True, #
ignore_constraints=False, #
method="jac",
dprocess=None,
interp_method="cubic",
mu=2,
maxbsteps=10,
tol=1e-8,
smaxit=500,
maxit=1000,
compute_radius=False,
invmethod="iti",
# obsolete
complementarities=None
) -> ImprovedTimeIterationResult:
# obsolete
if complementarities is not None:
# TODO: warning
pass
else:
complementarities = not ignore_constraints
def vprint(*args, **kwargs):
if verbose:
print(*args, **kwargs)
itprint = IterationsPrinter(
("N", int),
("f_x", float),
("d_x", float),
("Time_residuals", float),
("Time_inversion", float),
("Time_search", float),
("Lambda_0", float),
("N_invert", int),
("N_search", int),
verbose=verbose,
)
itprint.print_header("Start Improved Time Iterations.")
f = model.functions["arbitrage"]
g = model.functions["transition"]
x_lb = model.functions["arbitrage_lb"]
x_ub = model.functions["arbitrage_ub"]
parms = model.calibration["parameters"]
grid, dprocess_ = model.discretize()
if dprocess is None:
dprocess = dprocess_
endo_grid = grid["endo"]
exo_grid = grid["exo"]
n_m = max(dprocess.n_nodes, 1)
n_s = len(model.symbols["states"])
if interp_method in ("cubic", "linear"):
ddr = DecisionRule(
exo_grid, endo_grid, dprocess=dprocess, interp_method=interp_method
)
ddr_filt = DecisionRule(
exo_grid, endo_grid, dprocess=dprocess, interp_method=interp_method
)
else:
raise Exception("Unsupported interpolation method.")
# s = ddr.endo_grid
s = endo_grid.nodes
N = s.shape[0]
n_x = len(model.symbols["controls"])
x0 = (
model.calibration["controls"][
None,
None,
]
.repeat(n_m, axis=0)
.repeat(N, axis=1)
)
if dr0 is not None:
for i_m in range(n_m):
x0[i_m, :, :] = dr0.eval_is(i_m, s)
ddr.set_values(x0)
steps = 0.5 ** numpy.arange(maxbsteps)
lb = x0.copy()
ub = x0.copy()
for i_m in range(n_m):
m = dprocess.node(i_m)
lb[i_m, :] = x_lb(m, s, parms)
ub[i_m, :] = x_ub(m, s, parms)
x = x0
# both affect the precision
ddr.set_values(x)
## memory allocation
n_im = dprocess.n_inodes(0) # we assume it is constant for now
jres = numpy.zeros((n_m, n_im, N, n_x, n_x))
S_ij = numpy.zeros((n_m, n_im, N, n_s))
for it in range(maxit):
jres[...] = 0.0
S_ij[...] = 0.0
t1 = time.time()
# compute derivatives and residuals:
# res: residuals
# dres: derivatives w.r.t. x
# jres: derivatives w.r.t. ~x
# fut_S: future states
ddr.set_values(x)
#
# ub[ub>100000] = 100000
# lb[lb<-100000] = -100000
#
# sh_x = x.shape
# rr =euler_residuals(f,g,s,x,ddr,dp,parms, diff=False, with_jres=False,set_dr=True)
# print(rr.shape)
#
# from iti.fb import smooth_
# jj = smooth_(rr, x, lb, ub)
#
# print("Errors with complementarities")
# print(abs(jj.max()))
#
# exit()
#
from dolo.numeric.optimize.newton import SerialDifferentiableFunction
sh_x = x.shape
ff = SerialDifferentiableFunction(
lambda u: euler_residuals(
f,
g,
s,
u.reshape(sh_x),
ddr,
dprocess,
parms,
diff=False,
with_jres=False,
set_dr=False,
).reshape((-1, sh_x[2]))
)
res, dres = ff(x.reshape((-1, sh_x[2])))
res = res.reshape(sh_x)
dres = dres.reshape((sh_x[0], sh_x[1], sh_x[2], sh_x[2]))
junk, jres, fut_S = euler_residuals(
f,
g,
s,
x,
ddr,
dprocess,
parms,
diff=False,
with_jres=True,
set_dr=False,
jres=jres,
S_ij=S_ij,
)
# if there are complementerities, we modify derivatives
if complementarities:
res, dres, jres = smooth(res, dres, jres, x - lb)
res[...] *= -1
dres[...] *= -1
jres[...] *= -1
res, dres, jres = smooth(res, dres, jres, ub - x, pos=-1.0)
res[...] *= -1
dres[...] *= -1
jres[...] *= -1
err_0 = abs(res).max()
# premultiply by A
jres[...] *= -1.0
for i_m in range(n_m):
for j_m in range(n_im):
M = jres[i_m, j_m, :, :, :]
X = dres[i_m, :, :, :].copy()
sol = solve_tensor(X, M)
t2 = time.time()
# new version
if invmethod == "gmres":
ddx = solve_gu(dres.copy(), res.copy())
L = Operator(jres, fut_S, ddr_filt)
n0 = L.counter
L.addid = True
ttol = err_0 / 100
sol = scipy.sparse.linalg.gmres(
L, ddx.ravel(), tol=ttol
) # , maxiter=1, restart=smaxit)
lam0 = 0.01
nn = L.counter - n0
tot = sol[0].reshape(ddx.shape)
else:
# compute inversion
tot, nn, lam0 = invert_jac(
res,
dres,
jres,
fut_S,
ddr_filt,
tol=tol,
maxit=smaxit,
verbose=(verbose == "full"),
)
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,tol=tol,maxit=1000,verbose=(verbose=='full'),filt=ddr_filt)
# backsteps
t3 = time.time()
for i_bckstps, lam in enumerate(steps):
new_x = x - tot * lam
new_err = euler_residuals(
f, g, s, new_x, ddr, dprocess, parms, diff=False, set_dr=True
)
if complementarities:
new_err = smooth_nodiff(new_err, new_x - lb)
new_err = smooth_nodiff(-new_err, ub - new_x)
new_err = abs(new_err).max()
if new_err < err_0:
break
err_2 = abs(tot).max()
t4 = time.time()
itprint.print_iteration(
N=it,
f_x=err_0,
d_x=err_2,
Time_residuals=t2 - t1,
Time_inversion=t3 - t2,
Time_search=t4 - t3,
Lambda_0=lam0,
N_invert=nn,
N_search=i_bckstps,
)
if err_0 < tol:
break
x = new_x
ddr.set_values(x)
itprint.print_finished()
# if compute_radius:
# return ddx,L
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,ddr_filt,tol=tol,maxit=smaxit,verbose=(verbose=='full'))
# return ddr, lam, lam_max, lambdas
# else:
if not details:
return ddr
else:
ddx = solve_gu(dres.copy(), res.copy())
L = Operator(jres, fut_S, ddr_filt)
if compute_radius:
lam = scipy.sparse.linalg.eigs(L, k=1, return_eigenvectors=False)
lam = abs(lam[0])
else:
lam = np.nan
# lam, lam_max, lambdas = radius_jac(res,dres,jres,fut_S,ddr_filt,tol=tol,maxit=smaxit,verbose=(verbose=='full'))
return ImprovedTimeIterationResult(
ddr, it, err_0, err_2, err_0 < tol, complementarities, lam, None, L
)