def disp_polyhedron(A=None,
b=None,
points=None,
rays=None,
c=None,
obj_val=None,
opt=None,
loc=None):
if loc is None and opt is not None:
loc = opt
f = Figure()
f.add_polyhedron(p, label='Polyhedron $P$', color='red')
f.set_xlim([p.xlim[0], p.xlim[1] + 1])
f.set_ylim([p.ylim[0], p.ylim[1] + 2])
if c is not None and obj_val is not None:
f.add_line(c,
obj_val,
p.xlim + [0.2, 0.8],
p.ylim + [0.2, 1.8],
linestyle='dashed',
color='black',
label="Objective Function")
if opt is not None:
f.add_point(opt, 0.04, 'red')
f.add_text(loc, r'$x^* = %s$' % str(opt))
f.show()
def disp_relaxation(A, b, cuts=[], sol=None, disj=[]):
#TODO: Check sense of inequalities by looking explicitly at
# lp.constraintsUpper and lp.constraintsLower
p = Polyhedron2D(A=A, b=b)
f = Figure()
f.add_polyhedron(p, label='Polyhedron $P$')
f.set_xlim(p.xlim)
f.set_ylim(p.ylim)
pI = p.make_integer_hull()
f.add_polyhedron(pI,
show_int_points=True,
color='red',
linestyle='dashed',
label='Convex hull of integer points')
for (coeff, r) in cuts:
f.add_line(coeff, r, p.xlim, p.ylim, color='green', linestyle='dashed')
for (coeff, r) in disj:
f.add_line(coeff, r, p.xlim, p.ylim, color='red', linestyle='dashed')
f.add_line(coeff,
r + 1,
p.xlim,
p.ylim,
color='red',
linestyle='dashed')
if sol is not None:
f.add_point(sol, radius=.05)
f.show()
def disp_polyhedron(A = None, b = None, points = None, rays = None, c = None, obj_val = None,
opt = None, loc = None):
if loc is None and opt is not None:
loc = opt
f = Figure()
f.add_polyhedron(p, label = 'Polyhedron $P$', color = 'red')
f.set_xlim([p.xlim[0], p.xlim[1]+1])
f.set_ylim([p.ylim[0], p.ylim[1]+2])
if c is not None and obj_val is not None:
f.add_line(c, obj_val, p.xlim + [0.2, 0.8], p.ylim + [0.2, 1.8],
linestyle = 'dashed', color = 'black', label = "Objective Function")
if opt is not None:
f.add_point(opt, 0.04, 'red')
f.add_text(loc, r'$x^* = %s$' % str(opt))
f.show()
def disp_relaxation(A, b, cuts = [], sol = None):
#TODO: Check sense of inequalities by looking explicitly at
# lp.constraintsUpper and lp.constraintsLower
p = Polyhedron2D(A = A, b = b)
f = Figure()
f.add_polyhedron(p, label = 'Polyhedron $P$')
f.set_xlim([p.xlim[0], p.xlim[1]])
f.set_ylim([p.ylim[0], p.ylim[1]])
pI = p.make_integer_hull()
f.add_polyhedron(pI, show_int_points = True, color = 'red',
linestyle = 'dashed',
label = 'Convex hull of integer points')
for (coeff, r) in cuts:
f.add_line(coeff, r, p.xlim, p.ylim, color = 'green', linestyle = 'dashed')
if sol is not None:
f.add_point(sol, radius = .05)
f.show()
b1 = [8, -3, -9, -19, 12]
b2 = [-13, -1, 3, 27, 5, 20]
sense = ('Min', '<=')
integerIndices = [0, 1]
c = [1, 0]
cuts = None
rhs = None
obj_val = 2
p1 = Polyhedron2D(A=A1, b=b1)
p2 = Polyhedron2D(A=A2, b=b2)
sR = p2.make_integer_hull()
f = Figure()
f.add_polyhedron(p1,
label='$\mathcal{P}^{\,\prime}$',
color='blue',
show_int_points=True)
f.add_polyhedron(p2,
label='$\mathcal{P}^{\,\prime\prime}$',
color='black',
show_int_points=True)
f.add_polyhedron(
sR,
label=
'$\operatorname{conv}(\mathcal{P}^{\,\prime\prime} \cap \mathbb{Z}^{\,2})$',
color='red',
linestyle='dashed')
f.set_xlim(p2.xlim + [0, 1])
f.set_ylim(p2.ylim + [0, 1])
opt = [2, 3.5]
f.add_point(opt, .05, 'red')
5,
20
]
sense = ('Min', '<=')
integerIndices = [0, 1]
c = [1, 0]
cuts = None
rhs = None
obj_val = 2
p1 = Polyhedron2D(A = A1, b = b1)
p2 = Polyhedron2D(A = A2, b = b2)
sR = p2.make_integer_hull()
f = Figure()
f.add_polyhedron(p1, label = '$\mathcal{P}^{\,\prime}$', color = 'blue', show_int_points = True)
f.add_polyhedron(p2, label = '$\mathcal{P}^{\,\prime\prime}$', color = 'black', show_int_points = True)
f.add_polyhedron(sR, label = '$\operatorname{conv}(\mathcal{P}^{\,\prime\prime} \cap \mathbb{Z}^{\,2})$', color = 'red', linestyle = 'dashed')
f.set_xlim(p2.xlim + [0, 1])
f.set_ylim(p2.ylim + [0, 1])
opt = [2, 3.5]
f.add_point(opt, .05, 'red')
f.add_text([opt[0]-0.25, opt[1]], r'$x^*$')
if c is not None and obj_val is not None:
f.add_line(c, obj_val, p2.xlim, p2.ylim + [0.5, - 1.5],
linestyle = 'dashed', color = 'green', label = "Objective Function")
f.show()
cuts = [[-3, 1]]
rhs = [-5]
opt = [[2.42, 2.25]]
obj_val = [2.42]
class GenericSeparation(object):
def __init__(self, origProb, x0):
self.prob = origProb
self.iter = 1
self.x0 = x0
self.var = origProb.variablesDict()
self.dim = len(self.var)
self.currentXtremePoint = {}
self.solver = None
self.c = dict(list((v.name,c) for v,c in origProb.objective.iteritems()))
for v in self.var:
if v in self.c:
continue
else:
self.c[v] = 0.0
self.generate_separation_problem()
self.piCurrent = dict([(v.name, 0) for v in self.var.values()])
self.piPrevious = dict([(v.name, 0) for v in self.var.values()])
self.extremePoints = []
self.f = Figure()
def generate_separation_problem(self):
self.sepProb = LpProblem (name='separation '+self.prob.name, sense=LpMinimize)
self.pi = dict(list((v,LpVariable('pi'+v, None, None, LpContinuous,
None))
for v in self.var))
self.sepProb += lpSum(self.pi[v]*self.x0[v] for v in self.var)
def generate_xtreme_point(self):
obj = LpConstraintVar()
for v in self.prob.variables():
obj.addVariable(v, self.piCurrent[v.name])
self.prob.setObjective(obj)
self.prob.solve()
#solvers.COIN_CMD.solve(self.prob)
for v in self.var:
self.currentXtremePoint[v] = self.var[v].value()
if self.output == 1:
currentXtremePointList = self.currentXtremePoint.items()
currentXtremePointList.sort()
for v in currentXtremePointList:
print v[0]+'\t', v[1]
self.extremePoints.append(self.currentXtremePoint.values())
return self.prob.objective.value()
def add_inequality(self):
# change this, you should not access sense directly call a method
self.sepProb += lpSum (self.currentXtremePoint[v]*self.pi[v] for v in self.var) >= 1
def separate(self, output = False, p = None):
self.output = output
while True:
print 'Iteration ', self.iter
if self.generate_xtreme_point() >= 1-EPS:
break
self.add_inequality()
if self.output:
print "Separation problem solution status:", LpStatus[self.sepProb.solve()]
for v in self.var:
if self.pi[v].value() is not None:
print self.pi[v].name+'\t\t', self.pi[v].value()
else:
print self.pi[v].name+'\t\t', 0
self.piPrevious = deepcopy(self.piCurrent)
for v in self.var:
if self.pi[v].value() is not None:
self.piCurrent[v] = self.pi[v].value()
else:
self.piCurrent[v] = 0
self.iter += 1
if p is not None:
self.f.initialize()
self.f.add_polyhedron(p, label = 'Polyhedron P')
xList = (self.x0.values()[0], self.x0.values()[1])
if len(self.extremePoints) > 2:
pp = Polyhedron2D(points = self.extremePoints)
self.f.add_polyhedron(pp, color = 'red', linestyle = 'dashed',
label = 'Convex Hull of Generated Points')
elif len(self.extremePoints) == 1:
self.f.add_point(self.extremePoints[0], radius = 0.05,
color = 'green')
self.f.add_text(self.extremePoints[0][0]-0.5,
self.extremePoints[0][1]-0.08, '$x^0$')
else:
self.f.add_line_segment(self.extremePoints[0],
self.extremePoints[1],
color = 'red',
linestyle = 'dashed',
label = 'Convex Hull of Generated Points')
self.f.set_xlim(p.plot_min[0], p.plot_max[0])
self.f.set_ylim(p.plot_min[1], p.plot_max[1])
self.f.add_point(xList, radius = 0.05, color = 'red')
self.f.add_text(xList[0]-0.5, xList[1]-0.08, '$x^*$')
dList = (self.piCurrent.values()[0], self.piCurrent.values()[1])
self.f.add_line(dList, 1,
p.plot_max, p.plot_min, color = 'green',
linestyle = 'dashed', label = 'Separating Hyperplane')
self.f.show()
if self.output:
print self.sepProb.objective.value()
# add obj function
prob += lpSum(mip.c[i]*x[i] for i in range(len(mip.A[0])))
# add constraints to the prob
for i in range(len(mip.A)):
prob += mip.A[i][0]*x[0] + mip.A[i][1]*x[1] <= mip.b[i]
return prob, x, mip
if __name__ == '__main__':
display = True
prob, vars, mip = read_instance('MIP8')
if display:
p = Polyhedron2D(A = mip.A, b = mip.b)
f = Figure()
f.add_polyhedron(p, show_int_points = True, label = 'Polyhedron P')
f.set_xlim(p.plot_min[0], p.plot_max[0])
f.set_ylim(p.plot_min[1], p.plot_max[1])
f.add_point(mip.x_sep, radius = 0.05, color = 'red')
f.add_text(mip.x_sep[0]-0.5, mip.x_sep[1]-0.08, '$x^*$')
f.show()
# This is the point to be separated
x0 = {}
for index, value in enumerate(mip.x_sep):
x0[vars[index].name] = value
i = 0
ic = GenericSeparation(prob, x0)
ic.separate(output = True, p = p)
print 'separation problem objective value', ic.sepProb.objective.value()
class GenericSeparation(object):
def __init__(self, origProb, x0, pi_init=None):
if pi_init == None:
pi_init = [(v.name, 0) for v in self.var.values()]
self.prob = origProb
self.iter = 1
self.x0 = x0
self.var = origProb.variablesDict()
self.dim = len(self.var)
self.currentXtremePoint = {}
self.solver = None
self.c = dict(
list((v.name, c) for v, c in origProb.objective.iteritems()))
for v in self.var:
if v in self.c:
continue
else:
self.c[v] = 0.0
self.generate_separation_problem()
self.piCurrent = dict([(v.name, pi_init[v.name])
for v in self.var.values()])
self.piPrevious = dict([(v.name, 0) for v in self.var.values()])
self.extremePoints = []
self.f = Figure()
def generate_separation_problem(self):
self.sepProb = LpProblem(name='separation ' + self.prob.name,
sense=LpMinimize)
self.pi = dict(
list((v, LpVariable('pi' + v, None, None, LpContinuous, None))
for v in self.var))
self.sepProb += lpSum(self.pi[v] * self.x0[v] for v in self.var)
def generate_xtreme_point(self):
obj = LpConstraintVar()
for v in self.prob.variables():
obj.addVariable(v, self.piCurrent[v.name])
self.prob.setObjective(obj)
self.prob.solve()
#solvers.COIN_CMD.solve(self.prob)
for v in self.var:
self.currentXtremePoint[v] = self.var[v].value()
if self.output == 1:
currentXtremePointList = self.currentXtremePoint.items()
currentXtremePointList.sort()
for v in currentXtremePointList:
print v[0] + '\t', v[1]
self.extremePoints.append(self.currentXtremePoint.values())
return self.prob.objective.value()
def add_inequality(self):
# change this, you should not access sense directly call a method
self.sepProb += lpSum(self.currentXtremePoint[v] * self.pi[v]
for v in self.var) >= 1
def separate(self, output=False, p=None):
self.output = output
while True:
print 'Iteration ', self.iter
if self.generate_xtreme_point() >= 1 - EPS:
break
self.add_inequality()
if self.output:
print "Separation problem solution status:", LpStatus[
self.sepProb.solve()]
for v in self.var:
if self.pi[v].value() is not None:
print self.pi[v].name + '\t\t', self.pi[v].value()
else:
print self.pi[v].name + '\t\t', 0
self.piPrevious = deepcopy(self.piCurrent)
for v in self.var:
if self.pi[v].value() is not None:
self.piCurrent[v] = self.pi[v].value()
else:
self.piCurrent[v] = 0
self.iter += 1
if p is not None:
self.f.initialize()
self.f.add_polyhedron(p, label='Polyhedron P')
xList = (self.x0.values()[0], self.x0.values()[1])
if len(self.extremePoints) > 2:
pp = Polyhedron2D(points=self.extremePoints)
self.f.add_polyhedron(
pp,
color='red',
linestyle='dashed',
label='Convex Hull of Generated Points')
elif len(self.extremePoints) == 1:
self.f.add_point(self.extremePoints[0],
radius=0.05,
color='green')
self.f.add_text([
self.extremePoints[0][0] - 0.5,
self.extremePoints[0][1] - 0.08
], '$x^0$')
else:
self.f.add_line_segment(
self.extremePoints[0],
self.extremePoints[1],
color='red',
linestyle='dashed',
label='Convex Hull of Generated Points')
self.f.set_xlim(p.xlim)
self.f.set_ylim(p.ylim)
self.f.add_point(xList, radius=0.05, color='red')
self.f.add_text([xList[0] - 0.5, xList[1] - 0.08], '$x^*$')
dList = (self.piCurrent.values()[0],
self.piCurrent.values()[1])
self.f.add_line(dList,
1,
p.xlim,
p.ylim,
color='green',
linestyle='dashed',
label='Separating Hyperplane')
self.f.show()
if self.output:
print self.sepProb.objective.value()
prob += lpSum(mip.c[i] * x[i] for i in range(len(mip.A[0])))
# add constraints to the prob
for i in range(len(mip.A)):
prob += mip.A[i][0] * x[0] + mip.A[i][1] * x[1] <= mip.b[i]
return prob, x, mip
if __name__ == '__main__':
display = True
prob, vars, mip = read_instance(module_name='MIP8')
if display:
p = Polyhedron2D(A=mip.A, b=mip.b)
f = Figure()
f.add_polyhedron(p, show_int_points=True, label='Polyhedron P')
f.set_xlim(p.xlim)
f.set_ylim(p.ylim)
f.add_point(mip.x_sep, radius=0.05, color='red')
f.add_text([mip.x_sep[0] - 0.5, mip.x_sep[1] - 0.08], '$x^*$')
f.show()
# This is the point to be separated
x0 = {}
for index, value in enumerate(mip.x_sep):
x0[vars[index].name] = value
i = 0
ic = GenericSeparation(prob, x0, {'x_0': -1, 'x_1': 0})
ic.separate(output=True, p=p)
print 'separation problem objective value', ic.sepProb.objective.value()