将CVX半定规划问题转换为CVXPY时遇复数值问题求助

First, let's get to the root of your problem: CVXPY does not support optimizing complex-valued objective functions. Even though the trace of your Hermitian matrix f is real, if c[0] is complex, your objective c[0] - cvx.trace(f) becomes a complex expression, which CVXPY can't handle for maximization/minimization. This is the core reason your code fails when c contains complex numbers.

Key Fixes & Explanations #

• Ensure a Real-Valued Objective
  In most Fourier-related SDP problems, the objective should be a real scalar. If c[0] is a complex Fourier coefficient, you likely want to maximize the real part of your objective (since maximizing a complex number isn't well-defined). Modify your objective to explicitly take the real part of c[0]:

  obj = cvx.Maximize(cvx.real(c[0]) - cvx.trace(f))
  

  If c[0] is supposed to be real (e.g., the DC component of Fourier coefficients), double-check your data preprocessing to ensure it's cast to a real number.

• Check Constraint Compatibility
  Complex equality constraints are supported in CVXPY—they automatically split into real and imaginary part constraints. Since f is Hermitian, the elements referenced in your constraints can be complex, so matching them to complex values in c is valid.

• Use a Solver That Supports Complex SDPs
  Not all CVXPY solvers handle complex problems well. Stick to solvers like SCS or MOSEK (which requires a license but is highly optimized) for complex semidefinite programming. Specify the solver explicitly when calling solve():

  sol = prob.solve(solver=cvx.SCS)
  

Modified Working Code #

import cvxpy as cvx
import numpy as np

# Example complex-valued Fourier coefficients
c = [1 + 0j, 2 + 3j]
n = len(c)

# Create Hermitian optimization variable
f = cvx.Variable((n, n), hermitian=True)

# Build constraints
constraints = [f >> 0]
for k in range(1, n):
    indices = [(i * n) + i - (n - k) for i in range(n - k, n)]
    constraints += [cvx.sum(cvx.vec(f)[indices]) == c[n - k]]

# Real-valued objective function (critical fix)
obj = cvx.Maximize(cvx.real(c[0]) - cvx.trace(f))

# Solve with a complex-compatible solver
prob = cvx.Problem(obj, constraints)
sol = prob.solve(solver=cvx.SCS)

print("Optimal value:", sol)
print("Optimized f matrix:\n", f.value)

Why This Works #

• The objective is now strictly real, which aligns with CVXPY's optimization requirements.
• Complex constraints are handled correctly by splitting into real/imaginary components under the hood.
• Specifying a complex-aware solver ensures the problem is processed without compatibility errors.

内容的提问来源于stack exchange，提问作者Wulfsta