/
linearize_twoterm_posynomials.py
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/
linearize_twoterm_posynomials.py
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from builtins import range
from builtins import object
import numpy as np
import scipy.optimize as op
import os
from gpkit import Variable, Monomial, Posynomial
class LinearizeTwoTermPosynomials(object):
"""
Linearizes two term posynomials
"""
def __init__(self, p):
self.p = p
@staticmethod
def tangent_point_func(k, x, eps):
"""
the function used to calculate the tangent points
:param k: the point of tangency
:param x: the old intersection point
:param eps: the error
:return: the equation of the tangent line
"""
# warnings.simplefilter("ignore"): # this is making things slower
return np.log(1 + np.exp(x)) - eps - np.log(1 + np.exp(k)) - np.exp(k) * (x - k) / (1 + np.exp(k))
@staticmethod
def intersection_point_func(x, a, b, eps):
"""
the function used to calculate the intersection points
:param x: the break point to be solved for
:param a: the slope of the tangent line
:param b: the intercept of the tangent line
:param eps: the linearization error
:return: the break point equation
"""
return a * x + b - np.log(1 + np.exp(x)) + eps
@staticmethod
def iterate_linearization_coeff(r, eps):
"""
Finds the appropriate slopes, intercepts, tangency points, and intersection points for a given linearization
error and number of piecewise linear sections
:param r: the number of piecewise linear sections
:param eps: linearization error
:return: the slopes, intercepts, tangency points, and intersection points
"""
if r < 2:
raise Exception('The number of piece-wise sections should two or larger')
a, b = [], []
x_intersection = []
x_tangent = []
x_intersection.append(np.log(np.exp(eps) - 1))
i = 1
while i < r - 1:
x_old = x_intersection[i - 1]
try:
tangent_point = op.newton(LinearizeTwoTermPosynomials.tangent_point_func, x_old + 1, args=(x_old, eps))
slope = np.exp(tangent_point) / (1 + np.exp(tangent_point))
intercept = -np.exp(tangent_point) * tangent_point / (1 + np.exp(tangent_point)) + np.log(
1 + np.exp(tangent_point))
intersection_point = op.newton(LinearizeTwoTermPosynomials.intersection_point_func,
tangent_point + 1, args=(slope, intercept, eps))
except RuntimeError:
return i, a, b, x_tangent, x_intersection
x_tangent.append(tangent_point)
a.append(slope)
b.append(intercept)
x_intersection.append(intersection_point)
i += 1
return r, a, b, x_tangent, x_intersection
@staticmethod
def compute_linearization_coeff(r, tol):
"""
Calculates the slopes, intercepts, tangency points, intersection points, and linearization error for a given
number of piecewise-linear sections
:param r: the number of piecewise-linear sections
:param tol: tolerance of the linearization parameters
:return: slopes, intercepts, tangency points, intersection points, and linearization error
"""
a = None
b = None
x_tangent = None
x_intersection = None
eps = None
eps_min = 0
eps_max = np.log(2)
delta = 100
delta_old = 200
while delta > tol and delta != delta_old:
delta_old = delta
eps = (eps_max + eps_min) / 2
x_final_theoretical = -np.log(np.exp(eps) - 1)
number_of_actual_r, a, b, x_tangent, x_intersection = \
LinearizeTwoTermPosynomials.iterate_linearization_coeff(r, eps)
x_final_actual = x_intersection[-1]
if x_final_actual > x_final_theoretical or number_of_actual_r < r:
eps_max = eps
else:
eps_min = eps
delta = np.abs(x_final_actual - x_final_theoretical)
return a, b, x_tangent, x_intersection, eps
@staticmethod
def linearization_coeff(r):
"""
Reads the slopes, intercepts, tangency points, intersection points, and linearization error for a given number
of piecewise-linear sections from a text file
:param r: the number of piecewise-linear sections
:return: slopes, intercepts, tangency points, intersection points, and linearization error
"""
if r < 2:
raise Exception('The number of piece-wise sections should two or larger')
if r < 100:
linearization_data_file = open(os.path.dirname(__file__) + "/data/linearization_data.txt", "r")
for _ in range(r-2):
linearization_data_file.readline()
line = linearization_data_file.readline()
data = line.split(": ")
slopes = data[0].split(", ")[0:-1]
slopes = [float(item) for item in slopes]
intercepts = data[1].split(", ")[0:-1]
intercepts = [float(item) for item in intercepts]
x_tangent = data[2].split(", ")[0:-1]
x_tangent = [float(item) for item in x_tangent]
x_intersection = data[3].split(", ")[0:-1]
x_intersection = [float(item) for item in x_intersection]
eps = float(data[4])
linearization_data_file.close()
return slopes, intercepts, x_tangent, x_intersection, eps
else:
return LinearizeTwoTermPosynomials.compute_linearization_coeff(r, 2*np.finfo(float).eps)
def linearize(self, m, r):
"""
Approximates a two term posynomial constraint by upper and lower piecewise-linear constraints
:param m: the index of the constraint
:param r: the number of piecewise-linear sections`
:return: the deprived of data upper and lower constraints and the common data containing constraints
"""
if r < 2:
raise Exception('The number of piece-wise sections should be two or larger')
if len(self.p.exps) > 2:
raise Exception('The posynomial is larger than a two term posynomial')
if len(self.p.exps) < 2:
return [], [], [self.p <= 1]
a, b, _, _, eps = LinearizeTwoTermPosynomials.linearization_coeff(r)
data_constraints = []
w = Variable('w_%s' % m)
no_data_constraints_upper = [w * np.exp(eps) <= 1]
no_data_constraints_lower = [w <= 1]
first_monomial, second_monomial = self.p.chop()
data_constraints += [first_monomial <= w]
for i in range(r - 2):
data_constraints += [first_monomial ** a[r - 3 - i] *
second_monomial ** a[i] * np.exp(b[i]) <= w]
data_constraints += [second_monomial <= w]
return no_data_constraints_upper, no_data_constraints_lower, data_constraints
if __name__ == '__main__':
pass