def test_list5(self):
M = self._setup()
M.cc = ComplementarityList(rule=(complements(M.y + M.x3, M.x1 +
2 * M.x2 == i)
for i in range(3)))
self._test("list5", M)
def create_submodel_kkt_block(instance, submodel, deterministic,
fixed_upper_vars):
"""
Add optimality conditions for the submodel
This assumes that the original model has the form:
min c1*x + d1*y
A3*x <= b3
A1*x + B1*y <= b1
min c2*x + d2*y + x'*Q*y
A2*x + B2*y + x'*E2*y <= b2
y >= 0
NOTE THE VARIABLE BOUNDS!
"""
fixed_vars = {id(v) for v in fixed_upper_vars}
#
# Populate the block with the linear constraints.
# Note that we don't simply clone the current block.
# We need to collect a single set of equations that
# can be easily expressed.
#
d2 = {}
B2 = {}
vtmp = {}
utmp = {}
sids_set = set()
sids_list = []
#
block = Block(concrete=True)
block.u = VarList(
) # Note: Dual variables associated to bounds in primal problem
block.v = VarList(
) # Note: Dual variables associated to constraints in primal problem
block.c1 = ConstraintList()
block.c2 = ComplementarityList()
block.c3 = ComplementarityList()
#
# Collect submodel objective terms
#
# TODO: detect fixed variables
#
for odata in submodel.component_data_objects(Objective, active=True):
if odata.sense == maximize:
d_sense = -1
else:
d_sense = 1
#
# Iterate through the variables in the representation
#
o_terms = generate_standard_repn(odata.expr, compute_values=False)
#
# Linear terms
#
for i, var in enumerate(o_terms.linear_vars):
if id(var) in fixed_vars:
#
# Skip fixed upper variables
#
continue
#
# Store the coefficient for the variable. The coefficient is
# negated if the objective is maximized.
#
id_ = id(var)
d2[id_] = d_sense * o_terms.linear_coefs[i]
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
#
# Quadratic terms
#
for i, var in enumerate(o_terms.quadratic_vars):
if id(var[0]) in fixed_vars:
if id(var[1]) in fixed_vars:
#
# Skip fixed upper variables
#
continue
#
# Add the linear term
#
id_ = id(var[1])
d2[id_] = d2.get(
id_, 0) + d_sense * o_terms.quadratic_coefs[i] * var[0]
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
elif id(var[1]) in fixed_vars:
#
# Add the linear term
#
id_ = id(var[0])
d2[id_] = d2.get(
id_, 0) + d_sense * o_terms.quadratic_coefs[i] * var[1]
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
else:
raise RuntimeError(
"Cannot apply this transformation to a problem with \
quadratic terms where both variables are in the lower level.")
#
# Stop after the first objective
#
break
#
# Iterate through all lower level variables, adding dual variables
# and complementarity slackness conditions for y bound constraints
#
for vcomponent in instance.component_objects(Var, active=True):
for ndx in vcomponent:
if id(vcomponent[ndx]) in fixed_vars:
#
# Skip fixed upper variables
#
continue
#
# For each index, get the bounds for the variable
#
lb, ub = vcomponent[ndx].bounds
if not lb is None:
#
# Add the complementarity slackness condition for a lower bound
#
v = block.v.add()
block.c3.add(complements(vcomponent[ndx] >= lb, v >= 0))
else:
v = None
if not ub is None:
#
# Add the complementarity slackness condition for an upper bound
#
w = block.v.add()
vtmp[id(vcomponent[ndx])] = w
block.c3.add(complements(vcomponent[ndx] <= ub, w >= 0))
else:
w = None
if not (v is None and w is None):
#
# Record the variables for which complementarity slackness conditions
# were created.
#
id_ = id(vcomponent[ndx])
vtmp[id_] = (v, w)
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
#
# Iterate through all constraints, adding dual variables and
# complementary slackness conditions (for inequality constraints)
#
for cdata in submodel.component_data_objects(Constraint, active=True):
if cdata.equality:
# Don't add a complementary slackness condition for an equality constraint
u = block.u.add()
utmp[id(cdata)] = (None, u)
else:
if not cdata.lower is None:
#
# Add the complementarity slackness condition for a greater-than inequality
#
u = block.u.add()
block.c2.add(complements(-cdata.body <= -cdata.lower, u >= 0))
else:
u = None
if not cdata.upper is None:
#
# Add the complementarity slackness condition for a less-than inequality
#
w = block.u.add()
block.c2.add(complements(cdata.body <= cdata.upper, w >= 0))
else:
w = None
if not (u is None and w is None):
utmp[id(cdata)] = (u, w)
#
# Store the coefficients for the constraint variables that are not fixed
#
c_terms = generate_standard_repn(cdata.body, compute_values=False)
#
# Linear terms
#
for i, var in enumerate(c_terms.linear_vars):
if id(var) in fixed_vars:
continue
id_ = id(var)
B2.setdefault(id_, {}).setdefault(id(cdata),
c_terms.linear_coefs[i])
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
#
# Quadratic terms
#
for i, var in enumerate(c_terms.quadratic_vars):
if id(var[0]) in fixed_vars:
if id(var[1]) in fixed_vars:
continue
id_ = id(var[1])
if id_ in B2:
B2[id_][id(cdata)] = c_terms.quadratic_coefs[i] * var[0]
else:
B2.setdefault(id_, {}).setdefault(
id(cdata), c_terms.quadratic_coefs[i] * var[0])
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
elif id(var[1]) in fixed_vars:
id_ = id(var[0])
if id_ in B2:
B2[id_][id(cdata)] = c_terms.quadratic_coefs[i] * var[1]
else:
B2.setdefault(id_, {}).setdefault(
id(cdata), c_terms.quadratic_coefs[i] * var[1])
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
else:
raise RuntimeError(
"Cannot apply this transformation to a problem with \
quadratic terms where both variables are in the lower level.")
#
# Generate stationarity equations
#
tmp__ = (None, None)
for vid in sids_list:
exp = d2.get(vid, 0)
#
lb_dual, ub_dual = vtmp.get(vid, tmp__)
if vid in vtmp:
if not lb_dual is None:
exp -= lb_dual # dual for variable lower bound
if not ub_dual is None:
exp += ub_dual # dual for variable upper bound
#
B2_ = B2.get(vid, {})
utmp_keys = list(utmp.keys())
if deterministic:
utmp_keys.sort(key=lambda x: utmp[x][0].local_name\
if utmp[x][1] is None else utmp[x][1].local_name)
for uid in utmp_keys:
if uid in B2_:
lb_dual, ub_dual = utmp[uid]
if not lb_dual is None:
exp -= B2_[uid] * lb_dual
if not ub_dual is None:
exp += B2_[uid] * ub_dual
if type(exp) in six.integer_types or type(exp) is float:
# TODO: Annotate the model as unbounded
raise IOError("Unbounded variable without side constraints")
block.c1.add(exp == 0)
return block
def test_list1(self):
M = self._setup()
M.cc = ComplementarityList()
M.cc.add(complements(M.y + M.x3, M.x1 + 2 * M.x2 == 0))
M.cc.add(complements(M.y + M.x3, M.x1 + 2 * M.x2 == 2))
self._test("list1", M)
def _apply_solver(self):
start_time = time.time()
M=self.options.dual_bound
if not self.options.dual_bound:
M=1e6
print(f'Dual bound not specified, set to default {M}')
delta = self.options.delta
if not self.options.delta:
delta = 0.05 #What should default robustness delta be if not specified? Or should I raise an error?
print(f'Robustness parameter not specified, set to default {delta}')
# matrix representation for bilevel problem
matrix_repn = BilevelMatrixRepn(self._instance,standard_form=False)
# each lower-level problem
submodel = [block for block in self._instance.component_objects(SubModel)][0]
if len(submodel) != 1:
raise Exception('Problem encountered, this is not a valid bilevel model for the solver.')
self._instance.reclassify_component_type(submodel, Block)
#varref(submodel)
#dataref(submodel)
all_vars = {key: var for (key, var) in matrix_repn._all_vars.items()}
# get the variables that are fixed for the submodel (lower-level block)
fixed_vars = {key: var for (key, var) in matrix_repn._all_vars.items() if key in matrix_repn._fixed_var_ids[submodel.name]}
#Is there a way to get integer, continuous, etc for the upper level rather than lumping them all into fixed?
# continuous variables in SubModel
c_vars = {key: var for (key, var) in matrix_repn._all_vars.items() if key in matrix_repn._c_var_ids - fixed_vars.keys()}
# binary variables in SubModel SHOULD BE EMPTY FOR THIS SOLVER
b_vars = {key: var for (key, var) in matrix_repn._all_vars.items() if key in matrix_repn._b_var_ids - fixed_vars.keys()}
if len(b_vars)!= 0:
raise Exception('Problem encountered, this is not a valid bilevel model for the solver. Binary variables present!')
# integer variables in SubModel SHOULD BE EMPTY FOR THIS SOLVER
i_vars = {key: var for (key, var) in matrix_repn._all_vars.items() if key in matrix_repn._i_var_ids - fixed_vars.keys()}
if len(i_vars) != 0:
raise Exception('Problem encountered, this is not a valid bilevel model for the solver. Integer variables present!')
# get constraint information related to constraint id, sign, and rhs value
sub_cons = matrix_repn._cons_sense_rhs[submodel.name]
cons= matrix_repn._cons_sense_rhs[self._instance.name]
# construct the high-point problem (LL feasible, no LL objective)
# s0 <- solve the high-point
# if s0 infeasible then return high_point_infeasible
xfrm = TransformationFactory('pao.bilevel.highpoint')
xfrm.apply_to(self._instance)
#
# Solve with a specified solver
#
solver = self.options.solver
if not self.options.solver:
solver = 'gurobi'
for c in self._instance.component_objects(Block, descend_into=False):
if 'hp' in c.name:
#if '_hp' in c.name:
c.activate()
with pyomo.opt.SolverFactory(solver) as opt:
self.results.append(opt.solve(c,
tee=self._tee,
timelimit=self._timelimit))
_check_termination_condition(self.results[-1])
c.deactivate()
if self.options.do_print==True:
print('Solution to the Highpoint Relaxation')
for _, var in all_vars.items():
var.pprint()
# s1 <- solve the optimistic bilevel (linear/linear) problem (call solver3)
# if s1 infeasible then return optimistic_infeasible'
with pyomo.opt.SolverFactory('pao.bilevel.blp_global') as opt:
opt.options.solver = solver
self.results.append(opt.solve(self._instance,tee=self._tee,timelimit=self._timelimit))
_check_termination_condition(self.results[-1])
if self.options.do_print==True:
print('Solution to the Optimistic Bilevel')
for _, var in all_vars.items():
var.pprint()
#self._instance.pprint() #checking for active blocks left over from previous solves
# sk <- solve the dual adversarial problem
# if infeasible then return dual_adversarial_infeasible
# Collect the vertices solutions for the dual adversarial problem
#Collect up the matrix B and the vector d for use in all adversarial feasibility problems
n=len(c_vars.items())
m=len(sub_cons.items())
K=len(cons.items())
B=np.empty([m,n])
L=np.empty([K,1])
i=0
p=0
for _, var in c_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
B[:,i]=np.transpose(np.array(A))
i+=1
_ad_block_name='_adversarial'
self._instance.add_component(_ad_block_name, Block(Any))
_Vertices_name='_Vertices'
_Vertices_B_name='_VerticesB'
self._instance.add_component(_Vertices_name,Param(cons.keys()*NonNegativeIntegers*sub_cons.keys(),mutable=True))
Vertices=getattr(self._instance,_Vertices_name)
self._instance.add_component(_Vertices_B_name,Param(cons.keys()*NonNegativeIntegers,mutable=True))
VerticesB=getattr(self._instance,_Vertices_B_name)
adversarial=getattr(self._instance,_ad_block_name)
#Add Adversarial blocks
for _cidS, _ in cons.items(): # <for each constraint in the upper-level problem>
(_cid,_)=_cidS
ad=adversarial[_cid] #shorthand
ad.alpha=Var(sub_cons.keys(),within=NonNegativeReals) #sub_cons.keys() because it's a dual variable on the lower level constraints
ad.beta=Var(within=NonNegativeReals)
Hk=np.empty([n,1])
i=0
d=np.empty([n,1])
ad.cons=Constraint(c_vars.keys()) #B^Talpha+beta*d>= H_k, v-dimension constraints so index by c_vars
lhs_expr = {key: 0. for key in c_vars.keys()}
rhs_expr = {key: 0. for key in c_vars.keys()}
for _vid, var in c_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
coef = A #+ dot(A_q.toarray(), _fixed)
(C, C_q, C_constant) = matrix_repn.cost_vectors(submodel, var)
d[i,0]=float(C)
lhs_expr[_vid]=float(C)*ad.beta
(A,A_q,sign,b)=matrix_repn.coef_matrices(self._instance,var)
idx = list(cons.keys()).index(_cidS)
Hk[i,0]=A[idx]
i+=1
for _cid2 in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid2)
lhs_expr[_vid] += float(coef[idx])*ad.alpha[_cid2]
rhs_expr[_vid] = float(A[idx])
expr = lhs_expr[_vid] >= rhs_expr[_vid]
if not type(expr) is bool:
ad.cons[_vid] = expr
else:
ad.cons[_vid] = Constraint.Skip
ad.Obj=Objective(expr=0) #THIS IS A FEASIBILITY PROBLEM
with pyomo.opt.SolverFactory(solver) as opt:
self.results.append(opt.solve(ad,
tee=self._tee,
timelimit=self._timelimit))
_check_termination_condition(self.results[-1])
ad.deactivate()
Bd=np.hstack((np.transpose(B),d))
Eye=np.identity(m+1)
Bd=np.vstack((Bd,Eye))
Hk=np.vstack((Hk,np.zeros((m+1,1))))
mat=np.hstack((-Hk,Bd))
mat=cdd.Matrix(mat,number_type='float')
mat.rep_type=cdd.RepType.INEQUALITY
poly=cdd.Polyhedron(mat)
ext=poly.get_generators()
extreme=np.array(ext)
if self.options.do_print==True:
print(ext)
(s,t)=extreme.shape
l=1
for i in range(0,s):
j=1
if extreme[0,i]==1:
for _scid in sub_cons.keys():
#for j in range(1,t-1): #Need to loop over extreme 1 to t-1 and link those to the cons.keys for alpha?
Vertices[(_cidS,l,_scid)]=extreme[i,j] #Vertex l of the k-th polytope
j+=1
VerticesB[(_cidS,l)]=extreme[i,t-1]
l+=1
L[p,0]=l-1
p+=1
#vertex enumeration goes from 1 to L
# Solving the full problem sn0
_model_name = '_extended'
_model_name = unique_component_name(self._instance, _model_name)
xfrm = TransformationFactory('pao.bilevel.highpoint') #5.6a-c
kwds = {'submodel_name': _model_name}
xfrm.apply_to(self._instance, **kwds)
extended=getattr(self._instance,_model_name)
extended.sigma=Var(c_vars.keys(),within=NonNegativeReals,bounds=(0,M))
extended.lam=Var(sub_cons.keys(),within=NonNegativeReals,bounds=(0,M))
#5.d
extended.d = Constraint(c_vars.keys()) #indexed by lower level variables
d_expr= {key: 0. for key in c_vars.keys()}
for _vid, var in c_vars.items():
(C, C_q, C_constant) = matrix_repn.cost_vectors(submodel, var) #gets d_i
d_expr[_vid]+=float(C)
d_expr[_vid]=d_expr[_vid]-extended.sigma[_vid]
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
for _cid, _ in sub_cons.items():
idx = list(sub_cons.keys()).index(_cid)
d_expr[_vid]+=extended.lam[_cid]*float(A[idx])
expr = d_expr[_vid] == 0
if not type(expr) is bool:
extended.d[_vid] = expr
else:
extended.d[_vid] = Constraint.Skip
#5.e (Complementarity)
extended.e = ComplementarityList()
for _cid, _ in sub_cons.items():
idx=list(sub_cons.keys()).index(_cid)
expr=0
for _vid, var in fixed_vars.items(): #A_i*x
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
expr+=float(A[idx])*fixed_vars[_vid]
for _vid, var in c_vars.items(): #B_i*v
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
expr+=float(A[idx])*c_vars[_vid]
expr=expr-float(b[idx])
extended.e.add(complements(extended.lam[_cid] >= 0, expr <= 0))
#5.f (Complementarity)
extended.f = ComplementarityList()
for _vid,var in c_vars.items():
extended.f.add(complements(extended.sigma[_vid]>=0,var>=0))
#Replace 5.h-5.j with 5.7 Disjunction
extended.disjunction=Block(cons.keys()) #One disjunction per adversarial problem, one adversarial problem per upper level constraint
k=0
for _cidS,_ in cons.items():
idxS=list(cons.keys()).index(_cidS)
[_cid,sign]=_cidS
disjunction=extended.disjunction[_cidS] #shorthand
disjunction.Lset=RangeSet(1,L[k,0])
disjunction.disjuncts=Disjunct(disjunction.Lset)
for i in disjunction.Lset: #defining the L disjuncts
l_expr=0
for _vid, var in c_vars.items():
(C, C_q, C_constant) = matrix_repn.cost_vectors(submodel, var)
l_expr+=float(C)*var #d^Tv
l_expr+=delta
l_expr=VerticesB[(_cidS,i)]*l_expr #beta(d^Tv+delta)
for _cid, Scons in sub_cons.items(): #SUM over i to ml
Ax=0
idx=list(sub_cons.keys()).index(_cid)
for _vid, var in fixed_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
Ax += float(A[idx])*var
l_expr+=Vertices[(_cidS,i,_cid)]*(float(b[idx])-Ax)
r_expr=0
for _vid,var in fixed_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(self._instance, var) #get q and G
r_expr=r_expr-float(A[idxS])*var
r_expr+=float(b[idxS])
disjunction.disjuncts[i].cons=Constraint(expr= l_expr<=r_expr)
disjunction.seven=Disjunction(expr=[disjunction.disjuncts[i] for i in disjunction.Lset],xor=False)
k+=1
#extended.pprint()
TransformationFactory('mpec.simple_disjunction').apply_to(extended)
bigm = TransformationFactory('gdp.bigm')
bigm.apply_to(extended)
with pyomo.opt.SolverFactory(solver) as opt:
self.results.append(opt.solve(extended,
tee=self._tee,
timelimit=self._timelimit))
_check_termination_condition(self.results[-1])
# Return the sn0 solution
if self.options.do_print==True:
print('Robust Solution')
for _vid, _ in fixed_vars.items():
fixed_vars[_vid].pprint()
for _vid, _ in c_vars.items():
c_vars[_vid].pprint()
extended.lam.pprint()
extended.sigma.pprint()
stop_time = time.time()
self.wall_time = stop_time - start_time
return pyutilib.misc.Bunch(rc=getattr(opt, '_rc', None),
log=getattr(opt, '_log', None))
# ex1c.py
import pyomo.environ as pyo
from pyomo.mpec import ComplementarityList, complements
n = 5
model = pyo.ConcreteModel()
model.x = pyo.Var(range(1, n + 1))
model.f = pyo.Objective(expr=sum(i * (model.x[i] - 1)**2
for i in range(1, n + 1)))
def compl_(model):
yield complements(model.x[1] >= 0, model.x[2] >= 0)
yield complements(model.x[2] >= 0, model.x[3] >= 0)
yield complements(model.x[3] >= 0, model.x[4] >= 0)
yield complements(model.x[4] >= 0, model.x[5] >= 0)
model.compl = ComplementarityList(rule=compl_)
def _apply_solver(self):
self.results = []
start_time = time.time()
#
# Solve with a specified solver
#
solver = self.options.solver
if not self.options.solver:
solver = 'gurobi'
bigm = TransformationFactory('gdp.bigm')
# Step 1. Initialization
LB = -inf
UB = inf
theta = 0.
xi = 10e-3
k = 0
epsilon = 1e-4
M = 1e6
# matrix representation for bilevel problem
matrix_repn = BilevelMatrixRepn(self._instance)
# each lower-level problem
submodel = [
block for block in self._instance.component_objects(SubModel)
][0]
if len(submodel) != 1:
raise Exception(
'Problem encountered, this is not a valid bilevel model for the solver.'
)
self._instance.reclassify_component_type(submodel, Block)
varref(submodel)
dataref(submodel)
# all algorithm blocks
algorithm_blocks = list()
algorithm_blocks.append(submodel)
all_vars = {key: var for (key, var) in matrix_repn._all_vars.items()}
# for k,v in all_vars.items():
# if v.ub is None:
# v.setub(M)
# if v.lb is None:
# v.setlb(-M)
# get the variables that are fixed for the submodel (lower-level block)
fixed_vars = {
key: var
for (key, var) in matrix_repn._all_vars.items()
if key in matrix_repn._fixed_var_ids[submodel.name]
}
# continuous variables in SubModel
c_vars = {
key: var
for (key, var) in matrix_repn._all_vars.items()
if key in matrix_repn._c_var_ids - fixed_vars.keys()
}
# binary variables in SubModel
b_vars = {
key: var
for (key, var) in matrix_repn._all_vars.items()
if key in matrix_repn._b_var_ids - fixed_vars.keys()
}
# integer variables in SubModel
i_vars = {
key: var
for (key, var) in matrix_repn._all_vars.items()
if key in matrix_repn._i_var_ids - fixed_vars.keys()
}
# get constraint information related to constraint id, sign, and rhs value
sub_cons = matrix_repn._cons_sense_rhs[submodel.name]
# lower bounding block name -- unique naming convention
_lower_model_name = '_p9'
_lower_model_name = unique_component_name(self._instance,
_lower_model_name)
# upper bounding block name -- unique naming convention
_upper_model_name = '_p7'
_upper_model_name = unique_component_name(self._instance,
_upper_model_name)
while k <= self._k_max_iter:
# Step 2. Lower Bounding
# Solve problem (P5) master problem.
# This includes equations (53), (12), (13), (15), and (54)
# On iteration k = 0, (54) does not exist. Instead of implementing (54),
# this approach applies problem (P9) which incorporates KKT-based tightening
# constraints, and a projection and indicator constraint set.
lower_bounding_master = getattr(self._instance, _lower_model_name,
None)
if lower_bounding_master is None:
xfrm = TransformationFactory('pao.bilevel.highpoint')
kwds = {'submodel_name': _lower_model_name}
xfrm.apply_to(self._instance, **kwds)
lower_bounding_master = getattr(self._instance,
_lower_model_name)
algorithm_blocks.append(lower_bounding_master)
_c_var_bounds_rule = lambda m, k: c_vars[k].bounds
_c_var_init_rule = lambda m, k: (c_vars[k].lb + c_vars[k].ub
) / 2
lower_bounding_master._iter_c = Var(
Any, bounds=(0, M), within=Reals,
dense=False) # set (iter k, c_var_ids)
lower_bounding_master._iter_c_tilde = Var(
c_vars.keys(),
bounds=_c_var_bounds_rule,
initialize=_c_var_init_rule,
within=Reals) # set (iter k, c_var_ids)
lower_bounding_master._iter_b = Var(
Any, within=Binary, bounds=(0, M),
dense=False) # set (iter k, b_var_ids)
lower_bounding_master._iter_i = Var(
Any, within=Integers, bounds=(0, M),
dense=False) # set (iter k, i_var_ids)
lower_bounding_master._iter_pi_tilde = Var(
sub_cons.keys(),
bounds=(0, M)) # set (iter k, sub_cons_ids)
lower_bounding_master._iter_pi = Var(
Any, bounds=(0, M),
dense=False) # set (iter k, sub_cons_ids)
lower_bounding_master._iter_t = Var(
Any, bounds=(0, M),
dense=False) # set (iter k, sub_cons_ids)
lower_bounding_master._iter_lambda = Var(
Any, bounds=(0, M),
dense=False) # set (iter k, sub_cons_ids)
m = lower_bounding_master # shorthand reference to model
if k == 0:
# constraint (74)
lhs_expr = 0.
rhs_expr = 0.
for _vid, var in c_vars.items():
(C, C_q,
C_constant) = matrix_repn.cost_vectors(submodel, var)
coef = float(C) # + dot(C_q,_fixed))
ref = m._map[var]
lhs_expr += coef * ref
rhs_expr += coef * m._iter_c_tilde[_vid]
expr = lhs_expr >= rhs_expr
if not type(expr) is bool:
lower_bounding_master.KKT_tight1 = Constraint(
expr=lhs_expr >= rhs_expr)
# constraint (75a)
lower_bounding_master.KKT_tight2a = Constraint(sub_cons.keys())
lhs_expr_a = {key: 0. for key in sub_cons.keys()}
rhs_expr_a = {key: 0. for key in sub_cons.keys()}
for _vid, var in c_vars.items():
(A, A_q, sign,
b) = matrix_repn.coef_matrices(submodel, var)
coef = A #+ dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
lhs_expr_a[_cid] += float(
coef[idx]) * m._iter_c_tilde[_vid]
for var in {**b_vars, **i_vars, **fixed_vars}.values():
(A, A_q, sign,
b) = matrix_repn.coef_matrices(submodel, var)
coef = A #+ dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
ref = m._map[var]
rhs_expr_a[_cid] += -float(coef[idx]) * ref
for _cid, b in sub_cons.items():
(_, sign) = _cid
rhs_expr_a[_cid] += b
if sign == 'l' or sign == 'g->l':
expr = lhs_expr_a[_cid] <= rhs_expr_a[_cid]
if sign == 'e' or sign == 'g':
raise Exception(
'Problem encountered, this problem is not in standard form.'
)
if not type(expr) is bool:
lower_bounding_master.KKT_tight2a[_cid] = expr
else:
lower_bounding_master.KKT_tight2a[
_cid] = Constraint.Skip
# constraint (75b)
lower_bounding_master.KKT_tight2b = Constraint(c_vars.keys())
lhs_expr_b = {key: 0. for key in c_vars.keys()}
rhs_expr_b = {key: 0. for key in c_vars.keys()}
for _vid, var in c_vars.items():
(A, A_q, sign,
b) = matrix_repn.coef_matrices(submodel, var)
coef = A #+ dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
lhs_expr_b[_vid] += float(
coef[idx]) * m._iter_pi_tilde[_cid]
(C, C_q,
C_constant) = matrix_repn.cost_vectors(submodel, var)
rhs_expr_b[_vid] = float(C) #+ dot(C_q,_fixed))
expr = lhs_expr_b[_vid] >= rhs_expr_b[_vid]
if not type(expr) is bool:
lower_bounding_master.KKT_tight2b[_vid] = expr
else:
lower_bounding_master.KKT_tight2b[
_vid] = Constraint.Skip
# constraint (76a)
lower_bounding_master.KKT_tight3a = ComplementarityList()
for _vid in c_vars.keys():
lower_bounding_master.KKT_tight3a.add(
complements(m._iter_c_tilde[_vid] >= 0,
lhs_expr_b[_vid] - rhs_expr_b[_vid] >= 0))
# constraint (76b)
lower_bounding_master.KKT_tight3b = ComplementarityList()
for _cid in sub_cons.keys():
lower_bounding_master.KKT_tight3b.add(
complements(m._iter_pi_tilde[_cid] >= 0,
rhs_expr_a[_cid] - lhs_expr_a[_cid] >= 0))
# constraint (77a)
lower_bounding_master.KKT_tight4a = Constraint(c_vars.keys())
for _vid in c_vars.keys():
lower_bounding_master.KKT_tight4a[
_vid] = m._iter_c_tilde[_vid] >= 0
# constraint (77b)
lower_bounding_master.KKT_tight4b = Constraint(sub_cons.keys())
for _cid in sub_cons.keys():
lower_bounding_master.KKT_tight4b[
_cid] = m._iter_pi_tilde[_cid] >= 0
# solve the HPR and check the termination condition
lower_bounding_master.activate()
TransformationFactory('mpec.simple_disjunction').apply_to(
lower_bounding_master)
bigm.apply_to(lower_bounding_master)
with pyomo.opt.SolverFactory(solver) as opt:
self.results.append(
opt.solve(lower_bounding_master,
tee=self._tee,
timelimit=self._timelimit))
if not _check_termination_condition(self.results[-1]):
raise Exception(
'The lower-bounding master is infeasible or sub-optimal.')
# the LB should be a sequence of non-decreasing lower-bounds
_lb = [
value(odata)
for odata in self._instance.component_objects(Objective)
if odata.parent_block() == self._instance
][0]
if _lb < LB:
raise Exception(
'The lower-bound should be non-decreasing; a decreasing lower-bound indicates an algorithm issue.'
)
LB = max(LB, _lb)
lower_bounding_master.deactivate()
# Step 3. Termination
if UB - LB < xi:
# print(UB)
# print(LB)
if self._instance.solutions.solutions:
self._instance.solutions.select(0, ignore_fixed_vars=True)
stop_time = time.time()
self.wall_time = stop_time - start_time
self.results_obj = self._setup_results_obj()
return pyutilib.misc.Bunch(rc=getattr(opt, '_rc', None),
log=getattr(opt, '_log', None))
# fix the upper-level (master) variables to solve (P6) and (P7)
for key, var in fixed_vars.items():
var.fix(var.value)
# Step 4. Subproblem 1
# Solve problem (P6) lower-level problem for fixed upper-level optimal vars.
# In iteration k=0, this first subproblem is always feasible; furthermore, the
# optimal solution to (P5), alternatively (P9), will also be feasible to (P6).
with pyomo.opt.SolverFactory(solver) as _opt:
_results = _opt.solve(submodel,
tee=self._tee,
timelimit=self._timelimit)
if not _check_termination_condition(_results):
raise Exception(
'The lower-level subproblem with fixed upper-level variables is infeasible or sub-optimal.'
)
theta = [
value(odata) for odata in submodel.component_objects(Objective)
][0]
# solution for all variable values
_fixed = array(
[var.value for (key, var) in matrix_repn._all_vars.items()])
# temporary dictionary for b_vars
_b_vars_subproblem1 = dict()
for _vid, var in b_vars.items():
_b_vars_subproblem1[_vid] = var.value
# temporary dictionary for i_vars
_i_vars_subproblem1 = dict()
for _vid, var in i_vars.items():
_i_vars_subproblem1[_vid] = var.value
# Step 5. Subproblem 2
# Solve problem (P7) upper bounding problem for fixed upper-level optimal vars.
upper_bounding_subproblem = getattr(self._instance,
_upper_model_name, None)
if upper_bounding_subproblem is None:
xfrm = TransformationFactory('pao.bilevel.highpoint')
kwds = {'submodel_name': _upper_model_name}
xfrm.apply_to(self._instance, **kwds)
upper_bounding_subproblem = getattr(self._instance,
_upper_model_name)
algorithm_blocks.append(upper_bounding_subproblem)
for odata in upper_bounding_subproblem.component_objects(
Objective):
upper_bounding_subproblem.del_component(odata)
# solve for the master problem objective value for just the lower level variables
upper_bounding_subproblem.del_component('objective')
obj_constant = 0.
obj_expr = 0.
for var in {**c_vars, **b_vars, **i_vars}.values():
(C, C_q,
C_constant) = matrix_repn.cost_vectors(self._instance, var)
if obj_constant == 0. and C_constant != 0.:
obj_constant += C_constant # only add the constant once
obj_expr += float(C + dot(C_q, _fixed)) * var
upper_bounding_subproblem.objective = Objective(expr=obj_expr +
obj_constant)
# include lower bound constraint on the subproblem objective
upper_bounding_subproblem.del_component('theta_pareto')
sub_constant = 0.
sub_expr = 0.
for var in all_vars.values():
(C, C_q, C_constant) = matrix_repn.cost_vectors(submodel, var)
if sub_constant == 0. and C_constant != 0.:
sub_constant += C_constant # only add the constant once
sub_expr += float(C + dot(C_q, _fixed)) * var
upper_bounding_subproblem.theta_pareto = Constraint(
expr=sub_expr + sub_constant >= theta)
with pyomo.opt.SolverFactory(solver) as opt:
self.results.append(
opt.solve(upper_bounding_subproblem,
tee=self._tee,
timelimit=self._timelimit))
if _check_termination_condition(self.results[-1]):
# calculate new upper bound
obj_constant = 0.
obj_expr = 0.
for var in fixed_vars.values():
(C, C_q, C_constant) = matrix_repn.cost_vectors(
self._instance, var)
if obj_constant == 0. and C_constant != 0.:
obj_constant += C_constant # only add the constant once
obj_expr += float(C + dot(C_q, _fixed)) * var.value
# line 16 of decomposition algorithm
_ub = obj_expr + obj_constant + [
value(odata) for odata in
upper_bounding_subproblem.component_objects(Objective)
][0]
UB = min(UB, _ub)
# unfix the upper-level variables
for var in fixed_vars.values():
var.unfix()
# fix the solution for submodel binary variables
for _vid, var in b_vars.items():
m._iter_b[(k, _vid)].fix(var.value)
# fix the solution for submodel integer variables
for _vid, var in i_vars.items():
m._iter_i[(k, _vid)].fix(var.value)
else: # infeasible problem
# unfix the upper-level variables
for var in fixed_vars.values():
var.unfix()
# retrieve b_vars from temporary dictionary for subproblem1
for _vid, var in b_vars.items():
m._iter_b[(k, _vid)].fix(_b_vars_subproblem1[_vid])
# retrieve i_vars from temporary dictionary for subproblem1
for _vid, var in i_vars.items():
m._iter_i[(k, _vid)].fix(_i_vars_subproblem1[_vid])
# Step 6. Tightening the Master Problem
projections = getattr(lower_bounding_master, 'projections', None)
if projections is None:
lower_bounding_master.projections = Block(Any)
projections = lower_bounding_master.projections
# constraint (79)
projections[k].projection1 = Constraint(sub_cons.keys())
lhs_expr_proj = {key: 0. for key in sub_cons.keys()}
rhs_expr_proj = {key: 0. for key in sub_cons.keys()}
for _vid, var in c_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
coef = A + dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
lhs_expr_proj[_cid] += float(
coef[idx]) * m._iter_c[(k, _vid)]
for _vid, var in b_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
coef = A + dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
rhs_expr_proj[_cid] += -float(coef[idx]) * m._iter_b[
(k, _vid)]
for _vid, var in i_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
coef = A + dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
rhs_expr_proj[_cid] += -float(coef[idx]) * m._iter_i[
(k, _vid)]
for _vid, var in fixed_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
coef = A + dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
rhs_expr_proj[_cid] += -float(coef[idx]) * var
for _cid, b in sub_cons.items():
(id, sign) = _cid
rhs_expr_proj[_cid] += b
if sign == 'l' or sign == 'g->l':
projections[k].projection1[
_cid] = lhs_expr_proj[_cid] - m._iter_t[
(k, _cid)] <= rhs_expr_proj[_cid]
if sign == 'e' or sign == 'g':
raise Exception(
'Problem encountered, this problem is not in standard form.'
)
# constraint (80a)
projections[k].projection2a = Constraint(c_vars.keys())
for _vid in c_vars.keys():
projections[k].projection2a[_vid] = m._iter_c[(k, _vid)] >= 0
# constraint (80b)
projections[k].projection2b = Constraint(sub_cons.keys())
for _cid in sub_cons.keys():
projections[k].projection2b[_cid] = m._iter_t[(k, _cid)] >= 0
# constraint (82)
projections[k].projection3 = Block()
projections[k].projection3.indicator = Disjunct()
projections[k].projection3.block = Disjunct()
projections[k].projection3.disjunct = Disjunction(expr=[
projections[k].projection3.indicator,
projections[k].projection3.block
])
projections[k].projection3.indicator.cons_feas = Constraint(
expr=sum(m._iter_t[(k, _cid)]
for _cid in sub_cons.keys()) >= epsilon)
disjunction = projections[k].projection3.block
# constraint (82a)
lhs_constant = 0.
lhs_expr = 0.
rhs_constant = 0.
rhs_expr = 0.
for _vid, var in {**c_vars, **b_vars, **i_vars}.items():
(C, C_q, C_constant) = matrix_repn.cost_vectors(submodel, var)
lhs_expr += float(C + dot(C_q, _fixed)) * var
if var.is_continuous():
rhs_expr += float(C + dot(C_q, _fixed)) * m._iter_c[(k,
_vid)]
if var.is_binary():
rhs_expr += float(C + dot(C_q, _fixed)) * m._iter_b[(k,
_vid)]
if var.is_integer():
rhs_expr += float(C + dot(C_q, _fixed)) * m._iter_i[(k,
_vid)]
disjunction.projection3a = Constraint(
expr=lhs_expr + lhs_constant >= rhs_expr + rhs_constant)
# constraint (82b)
disjunction.projection3b = Constraint(sub_cons.keys())
for _cid in sub_cons.keys():
(_, sign) = _cid
if sign == 'l' or sign == 'g->l':
expr = lhs_expr_proj[_cid] <= rhs_expr_proj[_cid]
if sign == 'e' or sign == 'g':
raise Exception(
'Problem encountered, this problem is not in standard form.'
)
if not type(expr) is bool:
disjunction.projection3b[_cid] = expr
else:
disjunction.projection3b[_cid] = Constraint.Skip
# constraint (82c)
disjunction.projection3c = Constraint(c_vars.keys())
lhs_expr = {key: 0. for key in c_vars.keys()}
rhs_expr = {key: 0. for key in c_vars.keys()}
for _vid, var in c_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
coef = A + dot(A_q.toarray(), _fixed)
idx = list(sub_cons.keys()).index(_cid)
lhs_expr[_vid] += float(coef[idx]) * m._iter_pi[(k, _cid)]
(C, C_q, C_constant) = matrix_repn.cost_vectors(submodel, var)
rhs_expr[_vid] = float(C + dot(C_q, _fixed))
expr = lhs_expr[_vid] >= rhs_expr[_vid]
if not type(expr) is bool:
disjunction.projection3c[_vid] = expr
else:
disjunction.projection3c[_vid] = Constraint.Skip
# constraint (82d)
disjunction.projection3d = ComplementarityList()
for _vid in c_vars.keys():
disjunction.projection3d.add(
complements(m._iter_c[(k, _vid)] >= 0,
lhs_expr[_vid] - rhs_expr[_vid] >= 0))
# constraint (82e)
disjunction.projection3e = ComplementarityList()
for _cid in sub_cons.keys():
disjunction.projection3e.add(
complements(
m._iter_pi[(k, _cid)] >= 0,
rhs_expr_proj[_cid] - lhs_expr_proj[_cid] >= 0))
# constraint (82f)
disjunction.projection3f = Constraint(c_vars.keys())
for _vid in c_vars.keys():
disjunction.projection3f[_vid] = m._iter_c[(k, _vid)] >= 0
# constraint (82g)
disjunction.projection3g = Constraint(sub_cons.keys())
for _cid in sub_cons.keys():
disjunction.projection3g[_cid] = m._iter_pi[(k, _cid)] >= 0
# constraint (83a)
projections[k].projection4a = Constraint(c_vars.keys())
lhs_expr = {key: 0. for key in c_vars.keys()}
for _vid, var in c_vars.items():
(A, A_q, sign, b) = matrix_repn.coef_matrices(submodel, var)
coef = A + dot(A_q.toarray(), _fixed)
for _cid in sub_cons.keys():
idx = list(sub_cons.keys()).index(_cid)
lhs_expr[_vid] += float(
coef[idx]) * m._iter_lambda[(k, _cid)]
expr = lhs_expr[_vid] >= 0
if not type(expr) is bool:
projections[k].projection4a[_vid] = expr
else:
projections[k].projection4a[_vid] = Constraint.Skip
# constraint (83b)
projections[k].projection4b = ComplementarityList()
for _vid in c_vars.keys():
projections[k].projection4b.add(
complements(m._iter_c[(k, _vid)] >= 0,
lhs_expr[_vid] >= 0))
# constraint (84a)
projections[k].projection5a = Constraint(sub_cons.keys())
for _cid in sub_cons.keys():
projections[k].projection5a[_cid] = 1 - m._iter_lambda[
(k, _cid)] >= 0
# constraint (84b)
projections[k].projection5b = ComplementarityList()
for _cid in sub_cons.keys():
projections[k].projection5b.add(
complements(m._iter_t[(k, _cid)] >= 0,
1 - m._iter_lambda[(k, _cid)] >= 0))
# constraint (85)
projections[k].projection6 = ComplementarityList()
for _cid in sub_cons.keys():
projections[k].projection6.add(
complements(
m._iter_lambda[(k, _cid)] >= 0, rhs_expr_proj[_cid] -
lhs_expr_proj[_cid] + m._iter_t[(k, _cid)] >= 0))
k = k + 1
# Step 7. Loop
if UB - LB < xi:
# print(UB)
# print(LB)
if self._instance.solutions.solutions:
self._instance.solutions.select(0, ignore_fixed_vars=True)
stop_time = time.time()
self.wall_time = stop_time - start_time
self.results_obj = self._setup_results_obj()
return pyutilib.misc.Bunch(rc=getattr(opt, '_rc', None),
log=getattr(opt, '_log', None))
def _add_optimality_conditions(self, instance, submodel):
"""
Add optimality conditions for the submodel
This assumes that the original model has the form:
min c1*x + d1*y
A3*x <= b3
A1*x + B1*y <= b1
min c2*x + d2*y
y >= 0
A2*x + B2*y <= b2
NOTE THE VARIABLE BOUNDS!
"""
#
# Populate the block with the linear constraints. Note that we don't simply clone the
# current block. We need to collect a single set of equations that can be easily
# expressed.
#
d2 = {}
B2 = {}
vtmp = {}
utmp = {}
sids_set = set()
sids_list = []
#
block = Block(concrete=True)
block.u = VarList()
block.v = VarList()
block.c1 = ConstraintList()
block.c2 = ComplementarityList()
block.c3 = ComplementarityList()
#
# Collect submodel objective terms
#
for odata in submodel.component_data_objects(Objective, active=True):
if odata.sense == maximize:
d_sense = -1
else:
d_sense = 1
#
# Iterate through the variables in the canonical representation
#
o_terms = generate_canonical_repn(odata.expr, compute_values=False)
for i in range(len(o_terms.variables)):
var = o_terms.variables[i]
if var.parent_component().name in self._fixed_upper_vars:
#
# Skip fixed upper variables
#
continue
#
# Store the coefficient for the variable. The coefficient is
# negated if the objective is maximized.
#
id_ = id(var)
d2[id_] = d_sense * o_terms.linear[i]
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
# Stop after the first objective
break
#
# Iterate through all lower level variables, adding dual variables
# and complementarity slackness conditions for y bound constraints
#
for vcomponent in instance.component_objects(Var, active=True):
if vcomponent.name in self._fixed_upper_vars:
#
# Skip fixed upper variables
#
continue
for ndx in vcomponent:
#
# For each index, get the bounds for the variable
#
lb, ub = vcomponent[ndx].bounds
if not lb is None:
#
# Add the complementarity slackness condition for a lower bound
#
v = block.v.add()
block.c3.add(complements(vcomponent[ndx] >= lb, v >= 0))
else:
v = None
if not ub is None:
#
# Add the complementarity slackness condition for an upper bound
#
w = block.v.add()
vtmp[id(vcomponent[ndx])] = w
block.c3.add(complements(vcomponent[ndx] <= ub, w >= 0))
else:
w = None
if not (v is None and w is None):
#
# Record the variables for which complementarity slackness conditions
# were created.
#
id_ = id(vcomponent[ndx])
vtmp[id_] = (v, w)
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
#
# Iterate through all constraints, adding dual variables and
# complementary slackness conditions (for inequality constraints)
#
for cdata in submodel.component_data_objects(Constraint, active=True):
if cdata.equality:
# Don't add a complementary slackness condition for an equality constraint
u = block.u.add()
utmp[id(cdata)] = (None, u)
else:
if not cdata.lower is None:
#
# Add the complementarity slackness condition for a greater-than inequality
#
u = block.u.add()
block.c2.add(
complements(-cdata.body <= -cdata.lower, u >= 0))
else:
u = None
if not cdata.upper is None:
#
# Add the complementarity slackness condition for a less-than inequality
#
w = block.u.add()
block.c2.add(complements(cdata.body <= cdata.upper,
w >= 0))
else:
w = None
if not (u is None and w is None):
utmp[id(cdata)] = (u, w)
#
# Store the coefficients for the contraint variables that are not fixed
#
c_terms = generate_canonical_repn(cdata.body, compute_values=False)
for i in range(len(c_terms.variables)):
var = c_terms.variables[i]
if var.parent_component().name in self._fixed_upper_vars:
continue
id_ = id(var)
B2.setdefault(id_, {}).setdefault(id(cdata), c_terms.linear[i])
if not id_ in sids_set:
sids_set.add(id_)
sids_list.append(id_)
#
# Generate stationarity equations
#
tmp__ = (None, None)
for vid in sids_list:
exp = d2.get(vid, 0)
#
lb_dual, ub_dual = vtmp.get(vid, tmp__)
if vid in vtmp:
if not lb_dual is None:
exp -= lb_dual # dual for variable lower bound
if not ub_dual is None:
exp += ub_dual # dual for variable upper bound
#
B2_ = B2.get(vid, {})
utmp_keys = list(utmp.keys())
if self._deterministic:
utmp_keys.sort(key=lambda x: utmp[x][0].cname()
if utmp[x][1] is None else utmp[x][1].cname())
for uid in utmp_keys:
if uid in B2_:
lb_dual, ub_dual = utmp[uid]
if not lb_dual is None:
exp -= B2_[uid] * lb_dual
if not ub_dual is None:
exp += B2_[uid] * ub_dual
if type(exp) in six.integer_types or type(exp) is float:
# TODO: Annotate the model as unbounded
raise IOError("Unbounded variable without side constraints")
else:
block.c1.add(exp == 0)
#
# Return block
#
return block
def create_model_replacing_LL_with_kkt(repn):
"""
TODO - Document this transformation
"""
U = repn.U
LL = repn.U.LL
N = len(LL)
#
# Create Pyomo model
#
M = pe.ConcreteModel()
M.U = pe.Block()
M.L = pe.Block(range(N))
M.kkt = pe.Block(range(N))
# upper- and lower-level variables
pyomo_util.add_variables(M.U, U)
for i in range(N):
L = LL[i]
# lower-level variables
pyomo_util.add_variables(M.L[i], L)
# dual variables
M.kkt[i].lam = pe.Var(range(len(L.b))) # equality constraints
M.kkt[i].nu = pe.Var(range(len(L.x)),
within=pe.NonNegativeReals) # variable bounds
# objective
e = pyomo_util.dot(U.c[U], U.x, num=1) + U.d
for i in range(N):
L = LL[i]
e += pyomo_util.dot(U.c[L], L.x, num=1)
M.o = pe.Objective(expr=e)
# upper-level constraints
pyomo_util.add_linear_constraints(M.U, U.A, U, L, U.b, U.inequalities)
for i in range(N):
# lower-level constraints
L = LL[i]
pyomo_util.add_linear_constraints(M.L[i], L.A, U, L, L.b,
L.inequalities)
for i in range(N):
L = LL[i]
# stationarity
M.kkt[i].stationarity = pe.ConstraintList()
# L_A_L' * lam
L_A_L_T = L.A[L].transpose().todok()
X = pyomo_util.dot(L_A_L_T, M.kkt[i].lam)
if L.c[L] is not None:
for k in range(len(L.c[L])):
M.kkt[i].stationarity.add(L.c[L][k] + X[k] -
M.kkt[i].nu[k] == 0)
for i in range(N):
# complementarity slackness - variables
M.kkt[i].slackness = ComplementarityList()
for j in M.kkt[i].nu:
M.kkt[i].slackness.add(
complements(M.L[i].xR[j] >= 0, M.kkt[i].nu[j] >= 0))
return M