Complex-valued problemsYALMIP supports complex valued constraints for all solvers by automatically converting complex-valued problems to real-valued problems. To begin with, let us just define a simple linear complex problem to illustrate how complex variables and constraints are generated and interpreted.
To see how complex-valued constraints can be used in a more advanced setting, we solve the covariance estimation problem from the SeDuMi manual. The problem is to find a positive-definite Hermitian Toeplitz matrix Z such that the Frobenious norm of P-Z is minimized (P is a given complex matrix.) The matrix P is
We define a complex-valued Toeplitz matrix of the corresponding dimension
A complex Toeplitz matrix is not Hermitian, but we can make it Hermitian if we remove the imaginary part on the diagonal.
Minimizing the Frobenious norm of P-Z can be cast as minimizing the Euclidean norm of the vectorized difference P(:)-Z(:). By using a Schur complement, we see that this can be written as the following SDP.
The problem can be implemented more efficiently using a second order cone constraint.
...or by using a quadratic objective function
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