# Could anyone tell me how to skip this dcp rules?

I want to solve this qcqp problems:

\hat{\xi}_i are constants, Q is a symmetrical matrix but notnecessarily negtive semidefinite, \epsilon is a constant and \lambda_{max}(Q) means the largest eginevalue of Q, which is a constant too. The decision variables are \theta_i and \alpha. This is a qcqp problem with a compact feasible set but not convex. Therefore, what i do is relax it to the following problem:

R = \text{max}\,\, \frac{1}{N} \text{trace}(QX + 2 \Xi\Theta) +\alpha\lambda_{max}(Q) \\ \text{s.t.}\,\, \alpha \in R_{+} ,\Theta \in R^{m\times N},X \in R^{m\times m}\\ \frac{1}{N}\sum_{i=1}^N\parallel\Theta\parallel_F^2 +\alpha \leq \epsilon^2\\ X\succeq \Theta\Theta^T

\Theta is the decision variable matrix, \Xi is the constant matrix consist of \xi_i, X is an auxiliary variables.
Here is my question: when I use cvx to solve my problems, I construct constraint X-\Theta\Theta^T \succeq 0 , and this constraint actually violate DCP rules since DCP doesn’t allow the multiplication of two variables. How can I escape this rules? Or, what can I do to solve the origin problems because maye my relaxation problems is wrong.
Thank you very much for answering my question!

variables X(m,m) theta(m)
[X theta;theta' 1] == semidefinite(m+1)


Edit: I think X should be declared symmetric, although I’m not sure it’s necessary. So probably better to use

variable X(m,m) symmetric
variable theta(m)
[X theta;theta' 1] == semidefinite(m+1)

Thank you very much, bro