-
Notifications
You must be signed in to change notification settings - Fork 121
/
test_sdp.jl
311 lines (270 loc) · 11.4 KB
/
test_sdp.jl
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
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
# TODO: uncomment vexity checks once SDP on vars/constraints changes vexity of problem
@testset "SDP Atoms: $solver" for solver in solvers
if can_solve_sdp(solver)
@testset "sdp variables" begin
y = Variable((2,2), :Semidefinite)
p = minimize(y[1,1])
# @fact vexity(p) --> ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 0 atol=TOL
y = Variable((3,3), :Semidefinite)
p = minimize(y[1,1], y[2,2]==1)
# @fact vexity(p) --> ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 0 atol=TOL
# Solution is obtained as y[2,2] -> infinity
# This test fails on Mosek. See
# https://github.com/JuliaOpt/Mosek.jl/issues/29
# y = Variable((2, 2), :Semidefinite)
# p = minimize(y[1, 1], y[1, 2] == 1)
# # @fact vexity(p) --> ConvexVexity()
# solve!(p, solver)
# @fact p.optval --> roughly(0, TOL)
y = Semidefinite(3)
p = minimize(sum(diag(y)), y[1, 1] == 1)
# @fact vexity(p) --> ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 1 atol=TOL
y = Variable((3, 3), :Semidefinite)
p = minimize(tr(y), y[2,1]<=4, y[2,2]>=3)
# @fact vexity(p) --> ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 3 atol=TOL
x = Variable(Positive())
y = Semidefinite(3)
p = minimize(y[1, 2], y[2, 1] == 1)
# @fact vexity(p) --> ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 1 atol=TOL
end
@testset "sdp constraints" begin
# This test fails on Mosek
x = Variable(Positive())
y = Variable((3, 3))
p = minimize(x + y[1, 1], isposdef(y), x >= 1, y[2, 1] == 1)
# @fact vexity(p) --> ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 1 atol=TOL
end
@testset "nuclear norm atom" begin
y = Semidefinite(3)
p = minimize(nuclearnorm(y), y[2,1]<=4, y[2,2]>=3, y[3,3]<=2)
@test vexity(p) == ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 3 atol=TOL
@test evaluate(nuclearnorm(y)) ≈ 3 atol=TOL
end
@testset "operator norm atom" begin
y = Variable((3,3))
p = minimize(opnorm(y), y[2,1]<=4, y[2,2]>=3, sum(y)>=12)
@test vexity(p) == ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 4 atol=TOL
@test evaluate(opnorm(y)) ≈ 4 atol=TOL
end
@testset "sigma max atom" begin
y = Variable((3,3))
p = minimize(sigmamax(y), y[2,1]<=4, y[2,2]>=3, sum(y)>=12)
@test vexity(p) == ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 4 atol=TOL
@test evaluate(sigmamax(y)) ≈ 4 atol=TOL
end
@testset "lambda max atom" begin
y = Semidefinite(3)
p = minimize(lambdamax(y), y[1,1]>=4)
@test vexity(p) == ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 4 atol=TOL
@test evaluate(lambdamax(y)) ≈ 4 atol=TOL
end
@testset "lambda min atom" begin
y = Semidefinite(3)
p = maximize(lambdamin(y), tr(y)<=6)
@test vexity(p) == ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 2 atol=TOL
@test evaluate(lambdamin(y)) ≈ 2 atol=TOL
end
@testset "matrix frac atom" begin
x = [1, 2, 3]
P = Variable(3, 3)
p = minimize(matrixfrac(x, P), P <= 2*eye(3), P >= 0.5 * eye(3))
@test vexity(p) == ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 7 atol=TOL
@test (evaluate(matrixfrac(x, P)))[1] ≈ 7 atol=TOL
end
@testset "matrix frac atom both arguments variable" begin
x = Variable(3)
P = Variable(3, 3)
p = minimize(matrixfrac(x, P), lambdamax(P) <= 2, x[1] >= 1)
@test vexity(p) == ConvexVexity()
solve!(p, solver)
@test p.optval ≈ 0.5 atol=TOL
@test (evaluate(matrixfrac(x, P)))[1] ≈ 0.5 atol=TOL
end
@testset "sum largest eigs" begin
x = Semidefinite(3)
p = minimize(sumlargesteigs(x, 2), x >= 1)
solve!(p, solver)
@test p.optval ≈ 3 atol=TOL
@test evaluate(x) ≈ ones(3, 3) atol=TOL
x = Semidefinite(3)
p = minimize(sumlargesteigs(x, 2), [x[i,:] >= i for i=1:3]...)
solve!(p, solver)
@test p.optval ≈ 8.4853 atol=TOL
x1 = Semidefinite(3)
p1 = minimize(lambdamax(x1), x1[1,1]>=4)
solve!(p1, solver)
x2 = Semidefinite(3)
p2 = minimize(sumlargesteigs(x2, 1), x2[1,1]>=4)
solve!(p2, solver)
@test p1.optval ≈ p2.optval atol=TOL
x1 = Semidefinite(3)
p1 = minimize(lambdamax(x1), [x1[i,:] >= i for i=1:3]...)
solve!(p1, solver)
x2 = Semidefinite(3)
p2 = minimize(sumlargesteigs(x2, 1), [x2[i,:] >= i for i=1:3]...)
solve!(p2, solver)
@test p1.optval ≈ p2.optval atol=TOL
println(p1.optval)
end
@testset "kron atom" begin
id = eye(4)
X = Semidefinite(4)
W = kron(id, X)
p = maximize(tr(W), tr(X) ≤ 1)
@test vexity(p) == AffineVexity()
solve!(p, solver)
@test p.optval ≈ 4 atol=TOL
end
@testset "Partial trace" begin
A = Semidefinite(2)
B = [1 0; 0 0]
ρ = kron(B, A)
constraints = [partialtrace(ρ, 1, [2; 2]) == [0.09942819 0.29923607; 0.29923607 0.90057181], ρ in :SDP]
p = satisfy(constraints)
solve!(p, solver)
@test evaluate(ρ) ≈ [0.09942819 0.29923607 0 0; 0.299237 0.900572 0 0; 0 0 0 0; 0 0 0 0] atol=TOL
@test evaluate(partialtrace(ρ, 1, [2; 2])) ≈ [0.09942819 0.29923607; 0.29923607 0.90057181] atol=TOL
function rand_normalized(n)
A = 5*randn(n, n) + im*5*randn(n, n)
A / tr(A)
end
As = [ rand_normalized(3) for _ = 1:5]
Bs = [ rand_normalized(2) for _ = 1:5]
p = rand(5)
AB = sum(i -> p[i]*kron(As[i],Bs[i]), 1:5)
@test partialtrace(AB, 2, [3, 2]) ≈ sum( p .* As )
@test partialtrace(AB, 1, [3, 2]) ≈ sum( p .* Bs )
A, B, C = rand(5,5), rand(4,4), rand(3,3)
ABC = kron(kron(A, B), C)
@test kron(A,B)*tr(C) ≈ partialtrace(ABC, 3, [5, 4, 3])
# Test 281
A = rand(6,6)
expr = partialtrace(Constant(A), 1, [2, 3])
@test size(expr) == size(evaluate(expr))
@test_throws ArgumentError partialtrace(rand(6, 6), 3, [2, 3])
@test_throws ArgumentError partialtrace(rand(6, 6), 1, [2, 4])
@test_throws ArgumentError partialtrace(rand(3, 4), 1, [2, 3])
end
@testset "Optimization with complex variables" begin
@testset "Real Variables with complex equality constraints" begin
n = 10 # variable dimension (parameter)
m = 5 # number of constraints (parameter)
xo = rand(n)
A = randn(m,n) + im*randn(m,n)
b = A * xo
x = Variable(n)
p1 = minimize(sum(x), A*x == b, x>=0)
solve!(p1, solver)
x1 = x.value
p2 = minimize(sum(x), real(A)*x == real(b), imag(A)*x==imag(b), x>=0)
solve!(p2, solver)
x2 = x.value
@test x1 == x2
end
@testset "Complex Variable with complex equality constraints" begin
n = 10 # variable dimension (parameter)
m = 5 # number of constraints (parameter)
xo = rand(n)+im*rand(n)
A = randn(m,n) + im*randn(m,n)
b = A * xo
x = ComplexVariable(n)
p1 = minimize(real(sum(x)), A*x == b, real(x)>=0, imag(x)>=0)
solve!(p1, solver)
x1 = x.value
xr = Variable(n)
xi = Variable(n)
p2 = minimize(sum(xr), real(A)*xr-imag(A)*xi == real(b), imag(A)*xr+real(A)*xi == imag(b), xr>=0, xi>=0)
solve!(p2, solver)
#x2 = xr.value + im*xi.value
real_diff = real(x1) - xr.value
@test real_diff ≈ zeros(10, 1) atol=TOL
imag_diff = imag(x1) - xi.value
@test imag_diff ≈ zeros(10, 1) atol=TOL
#@fact x1==x2 --> true
end
@testset "norm2 atom" begin
a = 2+4im
x = ComplexVariable()
objective = norm2(a-x)
c1 = real(x)>=0
p = minimize(objective,c1)
solve!(p, solver)
@test p.optval ≈ 0 atol=TOL
@test evaluate(objective) ≈ 0 atol=TOL
real_diff = real(x.value) - real(a)
imag_diff = imag(x.value) - imag(a)
@test real_diff ≈ 0 atol=TOL
@test imag_diff ≈ 0 atol=TOL
end
@testset "sumsquares atom" begin
a = [2+4im;4+6im]
x = ComplexVariable(2)
objective = sumsquares(a-x)
c1 = real(x)>=0
p = minimize(objective,c1)
solve!(p, solver)
@test p.optval ≈ 0 atol=TOL
@test evaluate(objective) ≈ zeros(1, 1) atol=TOL
real_diff = real.(x.value) - real.(a)
imag_diff = imag.(x.value) - imag.(a)
@test real_diff ≈ zeros(2, 1) atol=TOL
@test imag_diff ≈ zeros(2, 1) atol=TOL
end
@testset "abs atom" begin
a = [5-4im]
x = ComplexVariable()
objective = abs(a-x)
c1 = real(x)>=0
p = minimize(objective,c1)
solve!(p, solver)
@test p.optval ≈ 0 atol=TOL
@test evaluate(objective) ≈ zeros(1) atol=TOL
real_diff = real(x.value) .- real(a)
imag_diff = imag(x.value) .- imag(a)
@test real_diff ≈ zeros(1) atol=TOL
@test imag_diff ≈ zeros(1) atol=TOL
end
@testset "Complex Semidefinite constraint" begin
n = 10
A = rand(n,n) + im*rand(n,n)
A = A + A' # now A is hermitian
x = ComplexVariable(n,n)
objective = sumsquares(A - x)
c1 = x in :SDP
p = minimize(objective, c1)
solve!(p, solver)
# test that X is approximately equal to posA:
l,v = eigen(A)
posA = v*Diagonal(max.(l,0))*v'
real_diff = real.(x.value) - real.(posA)
imag_diff = imag.(x.value) - imag.(posA)
@test real_diff ≈ zeros(n, n) atol=TOL
@test imag_diff ≈ zeros(n, n) atol=TOL
end
end
end
end