DDS Users' Guide

Epigraph of a matrix norm constraints

Without loss of generality, we can assume that our constraint is of the form \begin{eqnarray} && X-UU^\top \succeq 0, \nonumber \\ && X=A_0+\sum_{i=1}^\ell x_i A_i, \nonumber \\ && U=B_0+\sum_{i=1}^\ell x_i B_i, \end{eqnarray} where \(A_i\), \(i \in \{0,\ldots,\ell\}\), are \(m \times m\) symmetric matrices, and \(B_i\), \(i \in \{0,\ldots,\ell\}\), are \(m \times n\) matrices. We have two functions \(m2vec\) and \(vec2m\) for converting matrices (not necessarily symmetric) to vectors and vise versa. The abbreviation we use for epigraph of a matrix norm is MN. If we give these constraints as he \(k\)th block, cons{k,2} is a \(k\times 2\) matrix, where \(k\) is the number of constraints of this type, and each row is of the form \([m \ \ n]\). For each constraint of this form, the corresponding parts in \(A\) and \(b\) are \begin{eqnarray} \text{A\{k,1\}}=\left [\begin{array} {ccc} m2vec(B_1,n) & \cdots & m2vec(B_\ell,n) \\ sm2vec(A_1) & \cdots & sm2vec(A_\ell) \end{array} \right], \ \ \text{b\{k,1\}}=\left [\begin{array} {c} m2vec(B_0,n) \\ sm2vec(A_0) \end{array} \right]. \end{eqnarray}