I have formulated a Second-order Cone Problem (SOCP) in “quadratic” form with a norm inequality constraint. To use a certain solver (ECOS, to be precise), I need to rewrite it to a form that makes use of a generalized vector inequality over cones $K_i$.
My current problem is an SOCP of the following form:
$\text{minimize}\ f^T x $
$\text{subject to}\ ||A_i x + b_i|| \leq c_i^T x + d_i, i=1,2,...,m $
$ F x = g $
I now need to formulate it in this form with a generalized inequality:
$\text{minimize}\ f^T x $
$\text{subject to}\ Ax = b,\ Gx \leq_K h$
How should I approach this? Is there any reference that explains this? It would be great to get some references that could be useful for this.
You can find such notation and how it relates with the one you are used to in the classic book
Ben-Tal, Aharon, and Arkadi Nemirovski. Lectures on modern convex optimization: analysis, algorithms, and engineering applications. Vol. 2. Siam, 2001.
or more in brief in
http://docs.mosek.com/generic/modeling-a4.pdf
You basically introduce variables so that
$$ ||Ax - b|| \leq Cx +d $$
becomes
$$|| y || \leq z$$
$$z= Cx +d, y= Ax -b$$
and then you get $0\leq_K (y,z)$.