MATLAB: Nonconvex: Checking positive-semidefiniteness inside the Gloptipoly package

cvxgloptipolynonconvexpositive-semidefinite

Hi,

I’m dealing with a non-convex constraint and hence, instead of CVX (does CVX support MATLAB’s eig function?), I’m using Gloptipoly. When I use eig inside it, it throws up an error. Any alternative ways?

Thanks a lot.

Best Answer

For positive definiteness, I would recommended to check this using CHOL (see if the second output registers a problem), but this will not work for semi-definite. You could use LDL, and check if there are negative values in D.

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MATLAB: Chol() Error with Real, Symmetric, Positive Definite, 3-by-3 Matrix

Hi Matthias,

I understand that you are trying to use the 'chol' function to find the upper/lower triangular Cholesky factor of a certain 3x3 matrix. You are getting an error that the matrix 'must be positive definite', even though it looks like the matrix is already positive definite.

Unfortunately, it seems that the matrix X is not actually positive definite. To explain, the 'svd' function returns the singular values of the input matrix, not the eigenvalues. These two are not the same, and in particular, the singular values will always be nonnegative; therefore, they will not help in determining whether the eigenvalues are nonnegative. To obtain the eigenvalues, you can use the 'eig' function, as follows:

>> eig(X)
ans =
   -0.1885
    1.3032
    4.8853

Note that when you use the 'eig' function to obtain the eigenvalues, one of the values is -0.1885; therefore, the eigenvalues are not strictly positive, and the matrix is not positive definite.

For more information, see the documentation on singular values and eigenvalues:

http://www.mathworks.com/help/matlab/math/singular-values.html

http://www.mathworks.com/help/matlab/math/eigenvalues.html

MATLAB: How to determine if a matrix is positive definite using MATLAB

This change has been incorporated into the documentation in Release 14 Service Pack 3 (R14SP3). For previous releases, read below for any additional information:

Rather than using the EIG function to obtain the eigenvalues in order to determine positive definiteness, it is more computationally efficient to use the CHOL function. The CHOL function provides an optional second output argument "p" which is zero if the matrix is found to be positive definite. If the input matrix is not positive definite, then "p" will be a positive integer:

 >>[~,p] = chol(zeros(3))
 p =
        1

The CHOL function will return an error if it is only provided with a single output argument, and is also given a matrix that is not positive definite. NOTE: CHOL expects its input matrix to be symmetric and only looks at the upper triangular portion of the matrix.

Best Answer

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MATLAB: Chol() Error with Real, Symmetric, Positive Definite, 3-by-3 Matrix

MATLAB: How to determine if a matrix is positive definite using MATLAB

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