Turn non-convex constraint into a convex one

[shayan66 (shayan)]

Hi,

I hope everyone is doing well.

I ran into a problem! It is not a software issue. Assume that we have a constraint as (Convex>=Constant), which is a non-convex constraint.
I was wondering, is there any transformation/approximation I could do to turn this into a convex-constraint?
I do not want to use Taylor approximation since this specific constraint involves quadratic terms!

Methods like defining a slack variable or something else would be very intriguing to me.

Thanks a lot

[Mark_L_Stone]

No guarantees this will work well, or at all, on your problem.

How to handle nonlinear equality constraints?

mcg’s answer is correct; the problem is not convex, and cannot be expressed in DCP format. more specifically, a constraint like \|Aw\|_2 \leq P (with A a constant matrix and w a variable) is OK (convex, and DCP), but the converse constraint, \|Aw\|_2 \geq P, is not convex, and (therefore) not DCP. i’d like to add that there are heuristic methods for “handling” such problems, using the so-called convex-concave procedure (see EE364b, lecture on sequential convex programming). this is a heuris…

[shayan66 (shayan)]

Thank you so much for introducing this topic.

My problem is || W+Z ||_F^2 >= Constant (Frobenius norm of the sum of two matrices) where W and Z are both optimization variables. This is a non convex constraint.

I was wondering if there is any related topic to my problem that I can turn it into a convex constraint.

[Erling (Erling D.Andersen)]

What you ask is impossible.

[Mark_L_Stone]

As @Erling wrote, that is impossible. But you can try, with NO guarantees of success, to do something similar to what Stephen Boyd described in his linked (above) answer.

However, i suggest you use a non-convex solver to solve your problem.If the dimension is small enough, you might even succeed in globally optimizing it with a branch and bound global optimizer. YALMIP provides several solver options to handle such a problem, or you can call solvers such as FMINCON (local optimizer) or BARON (branch and bound global optimizer) directly from MATLAB.