# Figure 8.20, page 444.
# Floor planning example.
from cvxopt import solvers, matrix, spmatrix, mul, div
try: import pylab
except ImportError: pylab_installed = False
else: pylab_installed = True
def floorplan(Amin):
# minimize W+H
# subject to Amin1 / h1 <= w1
# Amin2 / h2 <= w2
# Amin3 / h3 <= w3
# Amin4 / h4 <= w4
# Amin5 / h5 <= w5
# x1 >= 0
# x2 >= 0
# x4 >= 0
# x1 + w1 + rho <= x3
# x2 + w2 + rho <= x3
# x3 + w3 + rho <= x5
# x4 + w4 + rho <= x5
# x5 + w5 <= W
# y2 >= 0
# y3 >= 0
# y5 >= 0
# y2 + h2 + rho <= y1
# y1 + h1 + rho <= y4
# y3 + h3 + rho <= y4
# y4 + h4 <= H
# y5 + h5 <= H
# h1/gamma <= w1 <= gamma*h1
# h2/gamma <= w2 <= gamma*h2
# h3/gamma <= w3 <= gamma*h3
# h4/gamma <= w4 <= gamma*h4
# h5/gamma <= w5 <= gamma*h5
#
# 22 Variables W, H, x (5), y (5), w (5), h (5).
#
# W, H: scalars; bounding box width and height
# x, y: 5-vectors; coordinates of bottom left corners of blocks
# w, h: 5-vectors; widths and heigths of the 5 blocks
rho, gamma = 1.0, 5.0 # min spacing, min aspect ratio
# The objective is to minimize W + H. There are five nonlinear
# constraints
#
# -w1 + Amin1 / h1 <= 0
# -w2 + Amin2 / h2 <= 0
# -w3 + Amin3 / h3 <= 0
# -w4 + Amin4 / h4 <= 0
# -w5 + Amin5 / h5 <= 0.
c = matrix(2*[1.0] + 20*[0.0])
def F(x=None, z=None):
if x is None:
return 5, matrix(17*[0.0] + 5*[1.0])
if min(x[17:]) <= 0.0:
return None
f = -x[12:17] + div(Amin, x[17:])
Df = matrix(0.0, (5,22))
Df[:,12:17] = spmatrix(-1.0, range(5), range(5))
Df[:,17:] = spmatrix(-div(Amin, x[17:]**2), range(5), range(5))
if z is None:
return f, Df
H = spmatrix( 2.0* mul(z, div(Amin, x[17::]**3)), range(17,22),
range(17,22) )
return f, Df, H
# linear inequalities
G = matrix(0.0, (26,22))
h = matrix(0.0, (26,1))
# -x1 <= 0
G[0,2] = -1.0
# -x2 <= 0
G[1,3] = -1.0
# -x4 <= 0
G[2,5] = -1.0
# x1 - x3 + w1 <= -rho
G[3, [2, 4, 12]], h[3] = [1.0, -1.0, 1.0], -rho
# x2 - x3 + w2 <= -rho
G[4, [3, 4, 13]], h[4] = [1.0, -1.0, 1.0], -rho
# x3 - x5 + w3 <= -rho
G[5, [4, 6, 14]], h[5] = [1.0, -1.0, 1.0], -rho
# x4 - x5 + w4 <= -rho
G[6, [5, 6, 15]], h[6] = [1.0, -1.0, 1.0], -rho
# -W + x5 + w5 <= 0
G[7, [0, 6, 16]] = -1.0, 1.0, 1.0
# -y2 <= 0
G[8,8] = -1.0
# -y3 <= 0
G[9,9] = -1.0
# -y5 <= 0
G[10,11] = -1.0
# -y1 + y2 + h2 <= -rho
G[11, [7, 8, 18]], h[11] = [-1.0, 1.0, 1.0], -rho
# y1 - y4 + h1 <= -rho
G[12, [7, 10, 17]], h[12] = [1.0, -1.0, 1.0], -rho
# y3 - y4 + h3 <= -rho
G[13, [9, 10, 19]], h[13] = [1.0, -1.0, 1.0], -rho
# -H + y4 + h4 <= 0
G[14, [1, 10, 20]] = -1.0, 1.0, 1.0
# -H + y5 + h5 <= 0
G[15, [1, 11, 21]] = -1.0, 1.0, 1.0
# -w1 + h1/gamma <= 0
G[16, [12, 17]] = -1.0, 1.0/gamma
# w1 - gamma * h1 <= 0
G[17, [12, 17]] = 1.0, -gamma
# -w2 + h2/gamma <= 0
G[18, [13, 18]] = -1.0, 1.0/gamma
# w2 - gamma * h2 <= 0
G[19, [13, 18]] = 1.0, -gamma
# -w3 + h3/gamma <= 0
G[20, [14, 18]] = -1.0, 1.0/gamma
# w3 - gamma * h3 <= 0
G[21, [14, 19]] = 1.0, -gamma
# -w4 + h4/gamma <= 0
G[22, [15, 19]] = -1.0, 1.0/gamma
# w4 - gamma * h4 <= 0
G[23, [15, 20]] = 1.0, -gamma
# -w5 + h5/gamma <= 0
G[24, [16, 21]] = -1.0, 1.0/gamma
# w5 - gamma * h5 <= 0.0
G[25, [16, 21]] = 1.0, -gamma
# solve and return W, H, x, y, w, h
sol = solvers.cpl(c, F, G, h)
return sol['x'][0], sol['x'][1], sol['x'][2:7], sol['x'][7:12], \
sol['x'][12:17], sol['x'][17:]
solvers.options['show_progress'] = False
if pylab_installed: pylab.figure(facecolor='w')
Amin = matrix([100., 100., 100., 100., 100.])
W, H, x, y, w, h = floorplan(Amin)
if pylab_installed:
if pylab_installed: pylab.subplot(221)
for k in range(5):
pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
[y[k], y[k]+h[k], y[k]+h[k], y[k]],
facecolor = '#D0D0D0')
pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
pylab.axis([-1.0, 26, -1.0, 26])
pylab.xticks([])
pylab.yticks([])
Amin = matrix([20., 50., 80., 150., 200.])
W, H, x, y, w, h = floorplan(Amin)
if pylab_installed:
pylab.subplot(222)
for k in range(5):
pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
[y[k], y[k]+h[k], y[k]+h[k], y[k]],
facecolor = '#D0D0D0')
pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
pylab.axis([-1.0, 26, -1.0, 26])
pylab.xticks([])
pylab.yticks([])
Amin = matrix([180., 80., 80., 80., 80.])
W, H, x, y, w, h = floorplan(Amin)
if pylab_installed:
pylab.subplot(223)
for k in range(5):
pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
[y[k], y[k]+h[k], y[k]+h[k], y[k]],
facecolor = '#D0D0D0')
pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
pylab.axis([-1.0, 26, -1.0, 26])
pylab.xticks([])
pylab.yticks([])
Amin = matrix([20., 150., 20., 200., 110.])
W, H, x, y, w, h = floorplan(Amin)
if pylab_installed:
pylab.subplot(224)
for k in range(5):
pylab.fill([x[k], x[k], x[k]+w[k], x[k]+w[k]],
[y[k], y[k]+h[k], y[k]+h[k], y[k]],
facecolor = '#D0D0D0')
pylab.text(x[k]+.5*w[k], y[k]+.5*h[k], "%d" %(k+1))
pylab.axis([-1.0, 26, -1.0, 26])
pylab.xticks([])
pylab.yticks([])
pylab.show()
