We know the standard form SDP has linear equality constraints (p.168 of Convex Optimization, S. Boyd):
In my study, I derive a trace product inequality like
$$\langle B_i,X\rangle \leq d_i\quad i=1\ldots n$$
where
- $X\in \mathbb{R^{n\times n}}$
- $B_i\in \mathbb{R^{n\times n}}$
- $d_i\in \mathbb{R}$
If I want to put this inequality in the original SDP problem, should I rewrite it as an equality constraint? or I can directly put it in the original SDP? Is there any suggested ways to deal with this?
You can put your inequality into the form of an equality by adding a slack variable and writing it as
$\langle B, X \rangle + s=d$
where
$s \geq 0$
To put this in matrix form, write your constraint as
$\langle A, Z \rangle = d$
where
$A=\left[ \begin{array}{cc} B & 0 \\ 0 & 1 \\ \end{array} \right]$
and
$Z=\left[ \begin{array}{cc} X & 0 \\ 0 & s \\ \end{array} \right]$.
Most software packages for SDP make it easy to add additional nonnegative variables to the problem as additional diagonal blocks without requiring extra storage for all of the off diagonal 0's.