# How to linearize this multiplicative constraint?

I have a constraint in the form

$$\sqrt{|\sum_{c\in C}{h_cw_c}|^2}\ge\sqrt{x}\zeta$$

Here, $$h_c$$ is s row vector (know), $$w_c$$ is a column vector (variable). $$x$$ and $$\zeta$$ are also optimization variables.

How can I transform this into linear/convex constraint?

Also, I have $$\text{imag}(h_cw_c)==0$$.

Of course, I can write the LHS part as $$||\sum_{c\in C}\rm{h_cw_c}||_2\ge\sqrt{x}\zeta$$.

Now, how to deal the RHS part?

Is there any way to linearize this?

• Are any of the variables binary? Otherwise, i think you're out of luck - face it, you have a non-convex nonlinear problem, and you're not going to cram this into CVX, if that is your hope, without a lot of bastardizations. – Mark L. Stone Dec 10 '20 at 13:07
• @MarkL.Stone, unfortunately..no. if not exact linearization, may be some convex overestimation is possible? – dipak narayanan Dec 10 '20 at 13:09
• Just use a non-convex solver. Try using YALMIP if you like the MATLAB environment. It handles problems CVX does in a similar, but not exactly the same, manner, and, unlike CVX, also can do non-convex problems and non-DCP convex problems, using either local or global optimization solver. The global solvers will do any under or overestimation for you, better than you would do on your own. – Mark L. Stone Dec 10 '20 at 13:11
• @MarkL.Stone, does YALMIP support binary variables? I have binary variable in other constraints of my problem? – dipak narayanan Dec 10 '20 at 13:14
• Yes, using solvers which natively support binary and integer, as well as its own top-level solvers, BNB (for convex problems) and BMIBNB for global solution of non-convex problems. You can use BMIBNB and specify FMINCON as upperesolver to globally optimize mixed-integer nonlinear non-convex problems (even though FMINCON does not support binary or integer problems, BMIBNB only feeds it continuous problems which it can handle. – Mark L. Stone Dec 10 '20 at 13:52