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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.

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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 $0\leq_K (y,z)$.

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  • $\begingroup$ Thanks, I'll see how I far I get from here! I'll accept the answer once I've solved my problem. $\endgroup$ – Ingo Mar 27 '14 at 9:34

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