Support more matrix norms
Description of possible additional matrix norms that could be supported:
- Induced norm
\|X\|_{\alpha,\beta}:=\max \{ \|Xu\|_{\beta}: \|u\|_{\alpha}\leq 1 \}
.
The constraint \|Xu\|_{\beta}\leq t
for all \|u\|_{\alpha}\leq 1
can be reformulated as a robust SOCP constraint subject to LP- \alpha
-ball uncertainty. If \alpha=2
this is a robust constraint with an Euclidean ball as uncertainty set, so this can be transformed to an SDP-constraint by using robust-picos. For other values of \alpha
, I don't know if this set is SDP-representable.
-
Schatten
$p$
-norm, i.e.,$p$
-norm of the vector of singular values. This is a convex, monotone, symmetrix, semidefinite representable function of the vector of singular values, so this can be handled via recipe in Proposition 3.2.2. of https://www2.isye.gatech.edu/~nemirovs/LMCO_LN.pdf -
Ky-Fan
$k$
-norms, i.e., sum of k largest singular values. Same as above.