1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
|
# The floor planning example section 9.2 (Problems with linear objectives).
from cvxopt import solvers, matrix, spmatrix, mul, div
solvers.options['show_progress'] = False
def floorplan(Amin):
# minimize W+H
# subject to Amink / hk <= wk, k = 1,..., 5
# 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
# hk/gamma <= wk <= gamma*hk, k = 1, ..., 5
#
# 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
#
# -wk + Amink / hk <= 0, k = 1, ..., 5
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
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:]
try:
import pylab
except ImportError:
pass
else:
pylab.figure(facecolor='w')
pylab.subplot(221)
Amin = matrix([100., 100., 100., 100., 100.])
W, H, x, y, w, h = floorplan(Amin)
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.subplot(222)
Amin = matrix([20., 50., 80., 150., 200.])
W, H, x, y, w, h = floorplan(Amin)
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.subplot(223)
Amin = matrix([180., 80., 80., 80., 80.])
W, H, x, y, w, h = floorplan(Amin)
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.subplot(224)
Amin = matrix([20., 150., 20., 200., 110.])
W, H, x, y, w, h = floorplan(Amin)
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()
|
