def impose_reweighted_variance(v, samples, weights=None, solver=None):
"""impose a variance on a list of points by reweighting weights"""
ndim = len(samples)
if weights is None:
weights = [1.0/ndim] * ndim
if solver is None or solver == 'fmin':
from mystic.solvers import fmin as solver
elif solver == 'fmin_powell':
from mystic.solvers import fmin_powell as solver
elif solver == 'diffev':
from mystic.solvers import diffev as solver
elif solver == 'diffev2':
from mystic.solvers import diffev2 as solver
norm = sum(weights)
m = mean(samples, weights)
inequality = ""
equality = ""; equality2 = ""; equality3 = ""
for i in range(ndim):
inequality += "x%s >= 0.0\n" % (i) # positive
equality += "x%s + " % (i) # normalized
equality2 += "%s * x%s + " % (float(samples[i]),(i)) # mean
equality3 += "x%s*(%s-%s)**2 + " % ((i),float(samples[i]),m) # var
equality += "0.0 = %s\n" % float(norm)
equality += equality2 + "0.0 = %s*%s\n" % (float(norm),m)
equality += equality3 + "0.0 = %s*%s\n" % (float(norm),v)
penalties = generate_penalty(generate_conditions(inequality))
constrain = generate_constraint(generate_solvers(solve(equality)))
def cost(x): return sum(x)
results = solver(cost, weights, constraints=constrain, \
penalty=penalties, disp=False, full_output=True)
wts = list(results[0])
_norm = results[1] # should have _norm == norm
warn = results[4] # nonzero if didn't converge
#XXX: better to fail immediately if xlo < m < xhi... or the below?
if warn or not almostEqual(_norm, norm):
print "Warning: could not impose mean through reweighting"
return None #impose_variance(v, samples, weights), weights
return wts #samples, wts # "mean-preserving"
return -(sin(2*pi*x0)**3 * sin(2*pi*x1)) / (x0**3 * (x0 + x1))
bounds = [(0,10)]*2
# with penalty='penalty' applied, solution is:
xs = [1.22797136, 4.24537337]
ys = -0.09582504
from mystic.symbolic import generate_constraint, generate_solvers, solve
from mystic.symbolic import generate_penalty, generate_conditions
equations = """
x0**2 - x1 + 1.0 <= 0.0
1.0 - x0 + (x1 - 4)**2 <= 0.0
"""
#cf = generate_constraint(generate_solvers(solve(equations))) #XXX: inequalities
pf = generate_penalty(generate_conditions(equations), k=1e12)
from mystic.constraints import as_constraint
cf = as_constraint(pf)
if __name__ == '__main__':
from mystic.solvers import buckshot
from mystic.math import almostEqual
result = buckshot(objective, 2, 40, bounds=bounds, penalty=pf, disp=False, full_output=True)
assert almostEqual(result[0], xs, tol=1e-2)
-x[0] + 2*x[1] <= 4
where: 0 <= x[0] <= inf
0 <= x[1] <= 4
"""
import numpy as np
import mystic.symbolic as ms
import mystic.solvers as my
import mystic.math as mm
# generate constraints and penalty for a linear system of equations
A = np.array([[1, 1],[-1, 2]])
b = np.array([7,4])
eqns = ms.linear_symbolic(G=A, h=b)
cons = ms.generate_constraint(ms.generate_solvers(ms.simplify(eqns)))
pens = ms.generate_penalty(ms.generate_conditions(eqns), k=1e3)
bounds = [(0., None), (0., 4.)]
# get the objective
def objective(x):
x = np.asarray(x)
return x[0]**2 + 4*x[1]**2 - 32*x[1] + 64
x0 = np.random.rand(2)
# compare against the exact minimum
xs = np.array([2., 3.])
ys = objective(xs)
sol = my.fmin_powell(objective, x0, constraint=cons, penalty=pens, disp=False,
def pf(len=3):
return generate_penalty(generate_conditions(equations(len)))
T + H + E + M + E - 72 == 0
V + I + O + L + I + N - 100 == 0
W + A + L + T + Z - 34 == 0
"""
var = list('ABCDEFGHIJKLMNOPQRSTUVWXYZ')
#NOTE: FOR A MORE DIFFICULT PROBLEM, COMMENT OUT THE FOLLOWING 5 LINES
bounds[0] = (5,5) # A
bounds[4] = (20,20) # E
bounds[8] = (25,25) # I
bounds[14] = (10,10) # O
bounds[20] = (1,1) # U
from mystic.constraints import unique, near_integers, has_unique
from mystic.symbolic import generate_penalty, generate_conditions
pf = generate_penalty(generate_conditions(equations,var),k=1)
from mystic.constraints import as_constraint
cf = as_constraint(pf)
from mystic.penalty import quadratic_equality
@quadratic_equality(near_integers)
@quadratic_equality(has_unique)
def penalty(x):
return pf(x)
from numpy import round, hstack, clip
def constraint(x):
x = round(x).astype(int) # force round and convert type to int
x = clip(x, 1,nletters) #XXX: hack to impose bounds
x = unique(x, list(range(1,nletters+1)))
return x
def objective(x):
x0, x1 = x
return 2 * x0**2 + x1**2 + x0 * x1 + x0 + x1
equations = """
x0 + x1 - 1.0 == 0.0
"""
bounds = [(0.0, None), (0.0, None)]
# with penalty='penalty' applied, solution is:
xs = [0.25, 0.75]
ys = 1.875
from mystic.symbolic import generate_conditions, generate_penalty
pf = generate_penalty(generate_conditions(equations), k=1e4)
from mystic.symbolic import generate_constraint, generate_solvers, solve
cf = generate_constraint(generate_solvers(solve(equations)))
if __name__ == '__main__':
from mystic.solvers import diffev2, fmin_powell
from mystic.math import almostEqual
result = diffev2(objective,
x0=bounds,
bounds=bounds,
constraint=cf,
penalty=pf,
npop=40,
disp=False,
T + H + E + M + E - 72 == 0
V + I + O + L + I + N - 100 == 0
W + A + L + T + Z - 34 == 0
"""
var = list('ABCDEFGHIJKLMNOPQRSTUVWXYZ')
#NOTE: FOR A MORE DIFFICULT PROBLEM, COMMENT OUT THE FOLLOWING 5 LINES
bounds[0] = (5, 5) # A
bounds[4] = (20, 20) # E
bounds[8] = (25, 25) # I
bounds[14] = (10, 10) # O
bounds[20] = (1, 1) # U
from mystic.constraints import unique, near_integers, has_unique
from mystic.symbolic import generate_penalty, generate_conditions
pf = generate_penalty(generate_conditions(equations, var), k=1)
from mystic.constraints import as_constraint
cf = as_constraint(pf)
from mystic.penalty import quadratic_equality
@quadratic_equality(near_integers)
@quadratic_equality(has_unique)
def penalty(x):
return pf(x)
from numpy import round, hstack, clip
def constraint(x):
var = list('ABC')
eqns = ms.simplify(equations, variables=var, all=True)
if isinstance(eqns, str):
_join = join_ = None
print(eqns)
else:
_join,join_ = _or,or_
for eqn in eqns:
print(eqn + '\n----------------------')
constrain = ms.generate_constraint(ms.generate_solvers(eqns, var, locals=dict(e_=eps)), join=_join)
solution = constrain([1,-2,1])
print('solved: %s' % dict(zip(var, solution)))
penalty = ms.generate_penalty(ms.generate_conditions(eqns, var, locals=dict(e_=eps)), join=join_)
print('penalty: %s' % penalty(solution))
equations = """
A*(B-C) + 2*C > 1
B + C = 2*D
D < 0
"""
print(equations)
var = list('ABCD')
eqns = ms.simplify(equations, variables=var, all=True)
if isinstance(eqns, str):
_join = join_ = None
print(eqns)
else:
_join,join_ = _or,or_
-x[0] + 2*x[1] <= 4
where: 0 <= x[0] <= inf
0 <= x[1] <= 4
"""
import numpy as np
import mystic.symbolic as ms
import mystic.solvers as my
import mystic.math as mm
# generate constraints and penalty for a linear system of equations
A = np.array([[1, 1], [-1, 2]])
b = np.array([7, 4])
eqns = ms.linear_symbolic(G=A, h=b)
cons = ms.generate_constraint(ms.generate_solvers(ms.simplify(eqns)))
pens = ms.generate_penalty(ms.generate_conditions(eqns), k=1e3)
bounds = [(0., None), (0., 4.)]
# get the objective
def objective(x):
x = np.asarray(x)
return x[0]**2 + 4 * x[1]**2 - 32 * x[1] + 64
x0 = np.random.rand(2)
# compare against the exact minimum
xs = np.array([2., 3.])
ys = objective(xs)
def pf(len=3):
return generate_penalty(generate_conditions(equations(len)))
