def interiorPath():
'''
Create a plot of a linear program together with the path followed
by the interior point algorithm.
'''
# create a simple linear program
A = np.array([[5., 1., -1, 0, 0], [1.5, 1., 0, -1, 0], [.2, 1., 0, 0, -1]])
b = np.array([3., 1.8, .75])
c = -np.array([3., 1., 0, 0, 0])
# initialize starting point
x, l, s = sol.startingPoint(-A, -b, c)
x += 3.5
s += 3.5
# solve the program
pts = sol.interiorPoint(-A,
-b,
-c,
niter=5,
starting_point=(x, l, s),
pts=True)
# plot the constraints together with the interior path
dom = np.linspace(0, 10, 2)
for i in xrange(3):
c = (-A[i, 0] * dom + b[i]) / A[i, 1]
plt.plot(dom, c, 'b')
pts = np.array(pts)[:, :2]
plt.plot(pts[:, 0], pts[:, 1], 'r*-')
plt.annotate(
'starting point',
xy=(pts[0, 0], pts[0, 1]),
xytext=(4, 3.5),
arrowprops=dict(facecolor='black', shrink=0.1),
)
plt.annotate(
'optimal point',
xy=(pts[-1, 0], pts[-1, 1]),
xytext=(1.5, 1.5),
arrowprops=dict(facecolor='black', shrink=0.1),
)
plt.text(2, 3, 'Feasible Region')
plt.ylim([0, 6])
plt.xlim([0, 6])
plt.savefig('interiorPath.pdf')
plt.clf()
def leastAbsDev():
"""
Plot a LAD and LSTSQ line for a set of data.
"""
# Generate some perturbed linear data
m = 10
n = 1
slope = 3.5
x = np.random.random(m).reshape((m, 1)) * 10
x = np.sort(x, axis=0)
y = slope * x + np.random.randn(m).reshape((m, 1))
y[-1] -= 20 # insert outlier
# Formulate constraint matrix
A = np.ones((2 * m, m + 2 + 2 * n + 2 * m))
A[::2, :m] = np.eye(m)
A[1::2, :m] = np.eye(m)
A[::2, m : m + n] = x
A[1::2, m : m + n] = -x
A[::2, m + n : m + 2 * n] = -x
A[1::2, m + n : m + 2 * n] = x
A[1::2, m + 2 * n] = -1
A[::2, m + 2 * n + 1] = -1
A[:, m + 2 + 2 * n :] = -np.eye(2 * m, 2 * m)
b = np.empty((2 * m, 1))
b[::2] = y
b[1::2] = -y
b = b.flatten()
c = np.zeros(A.shape[1])
c[:m] = 1
# Obtain and interpret solution
pts = sol.interiorPoint(A, b, c, niter=10, verbose=False)[0]
coeffs = pts[m : m + 2 * n + 2 : 2] - pts[m + 1 : m + 2 * n + 2 : 2]
# Obtain the least squares solution
B = np.ones((m, n + 1))
B[:, 0] = x.flatten()
coeffs2 = la.lstsq(B, y)[0]
# Plot the data, fitted lines
dom = np.linspace(0, 10, 2)
plt.subplot(211)
plt.scatter(x, y)
plt.plot(dom, coeffs[0] * dom + coeffs[1])
plt.subplot(212)
plt.scatter(x, y)
plt.plot(dom, coeffs2[0] * dom + coeffs2[1])
plt.savefig("leastAbsDev.pdf")
plt.clf()
def leastAbsDev():
"""
Plot a LAD and LSTSQ line for a set of data.
"""
#Generate some perturbed linear data
m = 10
n = 1
slope = 3.5
x = np.random.random(m).reshape((m, 1)) * 10
x = np.sort(x, axis=0)
y = slope * x + np.random.randn(m).reshape((m, 1))
y[-1] -= 20 #insert outlier
#Formulate constraint matrix
A = np.ones((2 * m, m + 2 + 2 * n + 2 * m))
A[::2, :m] = np.eye(m)
A[1::2, :m] = np.eye(m)
A[::2, m:m + n] = x
A[1::2, m:m + n] = -x
A[::2, m + n:m + 2 * n] = -x
A[1::2, m + n:m + 2 * n] = x
A[1::2, m + 2 * n] = -1
A[::2, m + 2 * n + 1] = -1
A[:, m + 2 + 2 * n:] = -np.eye(2 * m, 2 * m)
b = np.empty((2 * m, 1))
b[::2] = y
b[1::2] = -y
b = b.flatten()
c = np.zeros(A.shape[1])
c[:m] = 1
#Obtain and interpret solution
pts = sol.interiorPoint(A, b, c, niter=10, verbose=False)[0]
coeffs = pts[m:m + 2 * n + 2:2] - pts[m + 1:m + 2 * n + 2:2]
#Obtain the least squares solution
B = np.ones((m, n + 1))
B[:, 0] = x.flatten()
coeffs2 = la.lstsq(B, y)[0]
#Plot the data, fitted lines
dom = np.linspace(0, 10, 2)
plt.subplot(211)
plt.scatter(x, y)
plt.plot(dom, coeffs[0] * dom + coeffs[1])
plt.subplot(212)
plt.scatter(x, y)
plt.plot(dom, coeffs2[0] * dom + coeffs2[1])
plt.savefig('leastAbsDev.pdf')
plt.clf()
def interiorPath():
"""
Create a plot of a linear program together with the path followed
by the interior point algorithm.
"""
# create a simple linear program
A = np.array([[5.0, 1.0, -1, 0, 0], [1.5, 1.0, 0, -1, 0], [0.2, 1.0, 0, 0, -1]])
b = np.array([3.0, 1.8, 0.75])
c = -np.array([3.0, 1.0, 0, 0, 0])
# initialize starting point
x, l, s = sol.startingPoint(-A, -b, c)
x += 3.5
s += 3.5
# solve the program
pts = sol.interiorPoint(-A, -b, -c, niter=5, starting_point=(x, l, s), pts=True)
# plot the constraints together with the interior path
dom = np.linspace(0, 10, 2)
for i in xrange(3):
c = (-A[i, 0] * dom + b[i]) / A[i, 1]
plt.plot(dom, c, "b")
pts = np.array(pts)[:, :2]
plt.plot(pts[:, 0], pts[:, 1], "r*-")
plt.annotate(
"starting point", xy=(pts[0, 0], pts[0, 1]), xytext=(4, 3.5), arrowprops=dict(facecolor="black", shrink=0.1)
)
plt.annotate(
"optimal point", xy=(pts[-1, 0], pts[-1, 1]), xytext=(1.5, 1.5), arrowprops=dict(facecolor="black", shrink=0.1)
)
plt.text(2, 3, "Feasible Region")
plt.ylim([0, 6])
plt.xlim([0, 6])
plt.savefig("interiorPath.pdf")
plt.clf()