我正在尝试使用CVXPY软件包（与ECOS求解器）解决Python中的优化问题。问题涉及最小化受多个约束的线性变量。问题本身是拟凸的，因此我实现了对分算法以找到最佳解。
问题在于，当二等分越来越接近最佳值时，求解器总是会失败。就像求解器在解决方案接近最佳值时不知道该怎么做。我相信这个问题与公差问题有关，但是我不确定要更改哪个公差变量。
我尝试在Python 3.6中编码此问题。我也尝试过使用其他求解器，但是求解器在接近最优值时总是失败。代码如下所示。请注意，“ gamma”是我在此处最小化的变量。
import cvxpy as cp
import numpy as np
Ts = 500e-6;
w = np.logspace(-1, np.log10(np.pi/Ts), 400);
R = 0.1; L = 4e-2; a = R/L;
G = 0.0125/(np.exp(1j*w*Ts) - np.exp(-a*Ts))
bw = 200; zeta = 0.8;
wn = (2*np.pi*bw)/np.sqrt(1 - 2*np.power(zeta,2) + np.sqrt(2 - 4*np.power(zeta,2) + 4*np.power(zeta,4)))
Td = np.power(wn,2)/(-np.power(w,2) + 2*zeta*wn*1j*w + np.power(wn,2)); Wd = 1/(1-Td);
n = 5
rho_r = cp.Variable(n); rho_s = cp.Variable(n-1); rho_t = cp.Variable(n);
Ro = 0;
for i in range(n):
Ri = rho_r[i]*np.exp(-i*1j*w*Ts);
Ro = Ro + Ri;
So = 0
for i in range(n-1):
Si = rho_s[i]*np.exp(-(i+1)*1j*w*Ts);
So = So + Si;
So_no_int = 1+So
So = cp.multiply((1-np.exp(-1j*w*Ts)),(1+So));
To = 0;
for i in range(n):
Ti = rho_t[i]*np.exp(-i*1j*w*Ts);
To = To + Ti;
g_max = 2; g_min = 1e-8; g_tol = 1e-5; gamma = (g_max + g_min)/2;
mm = 0.5;
while g_max - g_min > g_tol:
PSI = So + cp.multiply(G,Ro)
F1 = cp.multiply(cp.inv_pos(gamma),cp.abs(cp.multiply(Wd,PSI - cp.multiply(G,To)))) - cp.real(PSI)
F2 = cp.abs(cp.multiply(mm,So)) - cp.real(PSI)
constraints = [F1 <= -1e-6, F2 <= -1e-6, cp.real(So_no_int) >= 5e-3]
prob = cp.Problem(cp.Minimize(0),constraints)
prob.solve(solver = cp.ECOS,feastol_inacc = 1e-7)
stat = prob.status
if stat == "optimal":
print("Feasible (gamma = ", gamma ,")")
gamma_opt = gamma;
rho_r_OPT = rho_r.value;
rho_s_OPT = rho_s.value;
rho_t_OPT = rho_t.value;
g_max = gamma;
gamma = np.average([gamma,g_min]);
GAIN = np.sum(rho_t_OPT)/np.sum(rho_r_OPT);
else:
print("Infeasible (gamma = ", gamma ,")")
g_min = gamma;
gamma = np.average([gamma,g_max]);
print("\nThe optimal solution gamma = %f" % gamma_opt)
这是输出。您可以看到解决方案收敛到接近1.059 ....的地方，但是求解器遇到错误。
Infeasible (gamma = 1.000000005 )
Feasible (gamma = 1.5000000025 )
Feasible (gamma = 1.2500000037499999 )
Feasible (gamma = 1.125000004375 )
Feasible (gamma = 1.0625000046875 )
Infeasible (gamma = 1.0312500048437498 )
Infeasible (gamma = 1.0468750047656248 )
Infeasible (gamma = 1.0546875047265623 )
Infeasible (gamma = 1.0585937547070312 )
Feasible (gamma = 1.0605468796972657 )
Feasible (gamma = 1.0595703172021484 )
Infeasible (gamma = 1.0590820359545898 )
Traceback (most recent call last):
File "X:\Google Drive\...
Stuff\...\Python_Controller_optimization\main_no_LMIs.py", line 60, in <module>
prob.solve(solver = cp.ECOS,feastol_inacc = 1e-7)
File "C:\Users\...\AppData\Local\Programs\Python\Python37-32\lib\site-packages\cvxpy\problems\problem.py", line 289, in solve
return solve_func(self, *args, **kwargs)
File "C:\Users\...\AppData\Local\Programs\Python\Python37-32\lib\site-packages\cvxpy\problems\problem.py", line 574, in _solve
self.unpack_results(solution, full_chain, inverse_data)
File "C:\Users\...\AppData\Local\Programs\Python\Python37-32\lib\site-packages\cvxpy\problems\problem.py", line 717, in unpack_results
"Try another solver, or solve with verbose=True for more "
cvxpy.error.SolverError: Solver 'ECOS' failed. Try another solver, or solve with verbose=True for more information.
[Finished in 4.8s]
以下是使用verbose = True进行的上等分迭代的结果：
ECOS 2.0.7 - (C) embotech GmbH, Zurich Switzerland, 2012-15. Web:
www.embotech.com/ECOS
It pcost dcost gap pres dres k/t mu step sigma IR | BT
0 +0.000e+00 -6.380e+02 +3e+03 1e-02 1e+01 1e+00 2e+00 --- --- 1 0 - | - -
1 +0.000e+00 -1.145e+02 +8e+02 4e-04 4e+00 1e+00 4e-01 0.8673 1e-01 1 1 1 | 0 0
2 +0.000e+00 -7.961e+01 +4e+02 3e-04 2e+00 6e-01 2e-01 0.7724 4e-01 2 1 2 | 0 0
3 +0.000e+00 -3.442e+01 +2e+02 1e-04 8e-01 2e-01 9e-02 0.6591 2e-01 2 1 2 | 0 0
4 +0.000e+00 -1.446e+01 +1e+02 6e-05 2e-01 5e-02 5e-02 0.7673 4e-01 2 2 2 | 0 0
5 +0.000e+00 -1.551e+00 +1e+01 6e-06 2e-02 4e-03 5e-03 0.8955 2e-02 2 1 1 | 0 0
6 +0.000e+00 -7.516e-01 +5e+00 3e-06 1e-02 1e-03 3e-03 0.7389 3e-01 2 2 2 | 0 0
7 +0.000e+00 -2.738e-01 +2e+00 1e-06 3e-03 5e-04 1e-03 0.7857 2e-01 3 3 3 | 0 0
8 +0.000e+00 -1.707e-01 +1e+00 7e-07 1e-03 3e-04 6e-04 0.7419 5e-01 3 3 3 | 0 0
9 +0.000e+00 -3.000e-02 +2e-01 1e-07 2e-04 4e-05 1e-04 0.8592 4e-02 2 3 3 | 0 0
10 +0.000e+00 -2.762e-02 +2e-01 1e-07 1e-04 4e-05 1e-04 0.1937 6e-01 3 3 4 | 0 0
11 +0.000e+00 -1.464e-02 +1e-01 6e-08 4e-05 2e-05 6e-05 0.6035 2e-01 3 4 4 | 0 0
12 +0.000e+00 -1.150e-02 +1e-01 5e-08 1e-05 2e-05 5e-05 0.5393 6e-01 3 4 4 | 0 0
13 +0.000e+00 -8.110e-03 +1e-01 3e-08 5e-06 1e-05 5e-05 0.4090 3e-01 4 5 5 | 0 0
14 +0.000e+00 -4.994e-03 +9e-02 2e-08 2e-06 8e-06 5e-05 0.5449 3e-01 5 6 6 | 0 0
15 +0.000e+00 -4.390e-03 +1e-01 2e-08 2e-06 8e-06 5e-05 0.2976 6e-01 5 6 6 | 0 0
16 +0.000e+00 -1.789e-03 +6e-02 8e-09 5e-07 4e-06 3e-05 0.6651 1e-01 4 5 5 | 0 0
17 +0.000e+00 -9.008e-04 +4e-02 5e-09 2e-07 2e-06 2e-05 0.5673 1e-01 5 6 6 | 0 0
18 +0.000e+00 -6.047e-05 +9e-03 3e-09 1e-08 4e-07 4e-06 0.9890 6e-02 5 5 6 | 0 0
19 +0.000e+00 -1.067e-05 +3e-03 3e-09 4e-09 3e-07 1e-06 0.9708 2e-01 5 6 6 | 0 0
20 +0.000e+00 -8.619e-07 +3e-04 3e-09 4e-09 3e-08 2e-07 0.9503 3e-02 6 6 6 | 0 0
21 +0.000e+00 -1.358e-07 +6e-05 3e-09 4e-09 6e-09 3e-08 0.8718 3e-02 6 5 6 | 0 0
22 +0.000e+00 -1.090e-07 +5e-05 3e-09 4e-09 7e-09 3e-08 0.4582 6e-01 5 5 5 | 0 0
23 +0.000e+00 -3.494e-08 +2e-05 3e-09 4e-09 4e-09 1e-08 0.8998 2e-01 7 5 5 | 0 0
24 +0.000e+00 -2.603e-08 +2e-05 3e-09 4e-09 3e-09 9e-09 0.5242 5e-01 7 5 5 | 0 0
25 +0.000e+00 -2.509e-08 +2e-05 3e-09 4e-09 3e-09 9e-09 0.1730 8e-01 7 5 5 | 0 0
26 +0.000e+00 -1.781e-08 +1e-05 3e-09 4e-09 3e-09 7e-09 0.4828 4e-01 5 5 5 | 0 0
27 +0.000e+00 -1.782e-08 +1e-05 3e-09 4e-09 3e-09 7e-09 0.0071 1e+00 6 5 5 | 0 0
28 +0.000e+00 -7.540e-09 +6e-06 3e-09 4e-09 2e-09 3e-09 0.9672 4e-01 5 5 6 | 0 0
29 +0.000e+00 -7.547e-09 +6e-06 3e-09 4e-09 2e-09 3e-09 0.0134 9e-01 5 5 5 | 0 0
30 +0.000e+00 -7.557e-09 +8e-06 1e-05 4e-09 2e-09 4e-09 0.1284 1e+00 0 4 6 | 0 0
Unreliable search direction detected, recovering best iterate (18) and stopping.
NUMERICAL PROBLEMS (reached feastol=1.1e-08, reltol=-nan(ind), abstol=8.9e-03).
Runtime: 0.182235 seconds.```
答案 0 :(得分：0)
问题可能在这里：
cp.multiply(cp.inv_pos(gamma),cp.abs(cp.multiply(Wd,PSI - cp.multiply(G,To)))) - cp.real(PSI)
cp.inv_pos(gamma)是一个凸的非负函数，cp.abs(...)也是一个凸函数。
当标量乘积的参数为非负数时，标量积是拟凹的，例如此处的情况。但是要满足准凹度，两个输入都必须是凹入的。由于输入为凸形，因此曲率未知。因此，问题本身是非凸的。 （请参见here。）