# How to deal with quadratic constrain in semidefinite programming

I am using CVX to solve an optimization problem. One of my constraints in the problem is

$$M \succeq \eta {\eta}^T$$

where $M$ is a square matrix and $\eta$ is a column vector (both $M$ and $\eta$ are variables). CVX issues the error

Only scalar quadratic forms can be specified in CVX.

I think the error results from the quadratic form of the constraint. I am wondering whether I can convert the constraint into another equivalent form?

(Update) Here is the optimization problem I am trying to solve:

\begin{align} \label{svm_new_obj_semi} & {\min}_{\boldsymbol{\eta}, M, \boldsymbol{\nu}, \boldsymbol{w}, \delta} \,\,\delta \\ \text{s.t.} & \quad \boldsymbol{\nu} \succeq 0 \nonumber \\ & \quad \boldsymbol{w} \succeq 0 \nonumber \\ & \quad 0 \preceq \boldsymbol{\eta} \preceq 1, \nonumber \\ & \quad M \succeq \boldsymbol{\eta} {\boldsymbol{\eta}}^T \nonumber \\ & \quad \begin{bmatrix} G\circ M & \boldsymbol{\eta} + \boldsymbol{\nu} - \boldsymbol{w} \\ {(\boldsymbol{\eta} + \boldsymbol{\nu} - \boldsymbol{w} )}^T & \frac{2}{\beta}( \delta - {\boldsymbol{w}}^T\boldsymbol{e} + {\boldsymbol{\eta}}^T \boldsymbol{e} ) \end{bmatrix} \succeq 0 \end{align}

I am trying to use: $0 \preceq M \preceq \boldsymbol{A}$, $diag(M) \preceq \boldsymbol{\eta}$, where $\boldsymbol{A}$ is a all ones matrix, to approximate $M \succeq \boldsymbol{\eta} {\boldsymbol{\eta}}^T$. Is this reasonable?

• What optimization problem? If $\eta$ does not show anywhere except in the constraint, then one can consider variable $\mathrm N \succeq \mathrm O$ instead of $\eta$, and then attempt to minimize its rank via minimization of the nuclear norm. Nov 8, 2016 at 17:37
• Thanks for your reply. The optimization problem is a semidefinite programming problem. I am trying to minimize $\delta$, which is the upper bound of my objective. $\eta$ only shows in constraints. $0 \preceq \eta \preceq 1$. Can you elaborate more on how to construct $N$? Thanks Nov 8, 2016 at 22:42
• @RodrigodeAzevedo I have posted my problem. Nov 8, 2016 at 22:55

Apply a Schur complement on the constraint $\begin{pmatrix} M & \eta\\\eta^T & 1\end{pmatrix} \succeq 0$