将半定程序从 CVX 转换为 CVXPY

Question

我想将以下 SDP（它只是验证约束的可行性）从 CVX (MATLAB) 转换为 CVXPY (Python)：

Ah = [1.0058, -0.0058; 1, 0];
Bh = [-1; 0];
Ch = [1.0058, -0.0058; -0.9829, 0.0056];
Dh = [-1; 1];

M = [0, 1;1, 0];
ni = size(M,1)/2;
n = size(Ah,1);
rho = 0.5;

cvx_begin sdp quiet
    variable P(n,n) semidefinite
    variable lambda(ni) nonnegative
    Mblk = M*kron(diag(lambda),eye(2));
    lambda(ni) == 1  % break homogeneity (many ways to do this...)
    [Ah Bh]'*P*[Ah Bh] - rho^2*blkdiag(P,0) + [Ch Dh]'*Mblk*[Ch Dh] <= 0
cvx_end


switch cvx_status
    case 'Solved'
        feas = 1;
    otherwise
        feas = 0;
end

下面是我的Python代码，

import cvxpy as cvx
import numpy as np
import scipy as sp


Ah = np.array([[1.0058, -0.0058], [1, 0]])
Bh = np.array([[-1], [0]])
Ch = np.array([[1.0058, -0.0058], [-0.9829, 0.0056]])
Dh = np.array([[-1], [1]])

M = np.array([[0, 1], [1, 0]])
ni, n = M.shape[0] / 2, Ah.shape[0]
rho = 0.5

P = cvx.Semidef(n)
lamda = cvx.Variable()

Mblk = np.dot(M, np.kron(cvx.diag(lamda), np.eye(2)))
ABh = np.concatenate((Ah, Bh), axis=1)
CDh = np.concatenate((Ch, Dh), axis=1)
constraints = [lamda[-1] == 1,
               np.dot(ABh.T, np.dot(P, ABh)) - rho**2*np.linalg.block_diag(P, 0) +
               np.dot(CDh.T, np.dot(Mblk, CDh)) << 0]

prob = cvx.Problem(cvx.Minimize(1), constraints)
feas = prob.status is cvx.OPTIMAL

我运行程序时有几个错误。 1.当我打印mblk时，它显示

Traceback (most recent call last):

File "/usr/lib/python2.7/dist-packages/IPython/core/interactiveshell.py", line 2820, in run_code

Out[1]: exec code_obj in self.user_global_ns, self.user_ns

File "", line 1, in

Mblk

File "/usr/lib/python2.7/dist-packages/IPython/core/displayhook.py", line 247, in call

format_dict, md_dict = self.compute_format_data(result)

File "/usr/lib/python2.7/dist-packages/IPython/core/displayhook.py", line 157, in compute_format_data

return self.shell.display_formatter.format(result)

File "/usr/lib/python2.7/dist-packages/IPython/core/formatters.py", line 152, in format

data = formatter(obj)

File "/usr/lib/python2.7/dist-packages/IPython/core/formatters.py", line 481, in call

printer.pretty(obj)

File "/usr/lib/python2.7/dist-packages/IPython/lib/pretty.py", line 362, in pretty

return _default_pprint(obj, self, cycle)

File "/usr/lib/python2.7/dist-packages/IPython/lib/pretty.py", line 482, in _default_pprint

p.text(repr(obj))

File "/usr/lib/python2.7/dist-packages/numpy/core/numeric.py", line 1553, in array_repr

', ', "array(")

File "/usr/lib/python2.7/dist-packages/numpy/core/arrayprint.py", line 454, in array2string

separator, prefix, formatter=formatter)

File "/usr/lib/python2.7/dist-packages/numpy/core/arrayprint.py", line 256, in _array2string

'int' : IntegerFormat(data),

File "/usr/lib/python2.7/dist-packages/numpy/core/arrayprint.py", line 641, in init

max_str_len = max(len(str(maximum.reduce(data))),

File "/usr/local/lib/python2.7/dist-packages/cvxpy/constraints/leq_constraint.py", line 67, in nonzero

Raise Exception("Cannot evaluate the truth value of a constraint.")

Exception: Cannot evaluate the truth value of a constraint.

当我走到这条线时，

  constraints = [lamda[-1] == 1,
                   np.dot(ABh.T, np.dot(P, ABh)) - rho**2*np.linalg.block_diag(P, 0) +
                   np.dot(CDh.T, np.dot(Mblk, CDh)) << 0]

显示

Traceback (most recent call last): File

".../sdp.py", line 22, in
np.dot(ABh.T, np.dot(P, ABh)) - rho**2*np.linalg.block_diag(P, 0) + 
ValueError: setting an array element with a sequence.

如何解决这些问题？

Answer 1

您的代码的最大问题是您不能在 CVXPY 对象上使用 NumPy 函数。您需要使用等效的 CVXPY 函数。这是您的代码的工作版本：

import cvxpy as cvx
import numpy as np
import scipy as sp


Ah = np.array([[1.0058, -0.0058], [1, 0]])
Bh = np.array([[-1], [0]])
Ch = np.array([[1.0058, -0.0058], [-0.9829, 0.0056]])
Dh = np.array([[-1], [1]])

M = np.array([[0, 1], [1, 0]])
ni, n = M.shape[0] / 2, Ah.shape[0]
rho = 0.5

P = cvx.Semidef(n)
lamda = cvx.Variable()

Mblk = M*lamda*np.eye(2)
ABh = cvx.hstack(Ah, Bh)
CDh = cvx.hstack(Ch, Dh)
zeros = np.zeros((n,1))
constraints = [lamda[-1] == 1,
               ABh.T*P*ABh - rho**2*cvx.bmat([[P,zeros],[zeros.T, 0]]) +
               CDh.T*Mblk*CDh << 0]

prob = cvx.Problem(cvx.Minimize(1), constraints)
prob.solve()
feas = prob.status is cvx.OPTIMAL

我删除了 kron 函数，因为它在这里没有做任何事情，而且 CVXPY 目前不支持左侧可变的 Kronecker 产品。有需要我可以加

将半定程序从 CVX 转换为 CVXPY

Convert a semidefinite program from CVX to CVXPY

python

matlab

convex-optimization

cvx

cvxpy