How should I write the following positive semidefinite constraint in cvx?

where, only \lambda_E, \xi and \alpha are variables. \boldsymbol{\rm A} has MxM dimensions, \boldsymbol{\rm a} has 1xM dimensions and a has 1x1 dimensions. How should I write the above positive semidefinite constraint in cvx? Thanks.

Unless there are some hidden relationships you haven’t told us about, this looks like a straightforward Linear SDP (LMI), which can be “directly” entered in CVX.

Have you read the CVX Users’ Guide, to include the Semidefinte programming Mode chapter?

After you have done so, if you still need help, please reply with the specifics of what you are having trouble with.

Hi Mark, this is my optimization problem

But I keep getting this error. Sometimes I get the optimum, but it is a negative number, and I want it to be positive.

This is my MATLAB code:

@tbchen wrote “it is a negative number, and I want it to be positive.”

I want a billion dollars. That doesn’t mean I’m going to get it.

Does your problem have a constraint that requires the optimum to be positive? CVX and Mosek aren’t mind readers, and won’t insert constraints not specified in the problem.

It appears you are using unsafeguarded (no line search or trust region) Scucessive Convex Approximation.

I will leave it to you to investigate whether the overall algorithm you are using might be easily repairable with the incorporation of an additional constraint in the sub-problems. I offer no opinion on the merits of whatever paper or book you found this algorithm in, or on the merits of the algorithms itself. But I will say as a general matter that there are SCA algorithms proposed in many papers published in even prestigious journals which are unreliable and which might not have passed my scrutiny had I been a referee or editor of the paper (I am tough, but fair)…