# 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. Dec 10, 2020 at 13:07
• @MarkL.Stone, unfortunately..no. if not exact linearization, may be some convex overestimation is possible?
– KGM
Dec 10, 2020 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. Dec 10, 2020 at 13:11
• @MarkL.Stone, does YALMIP support binary variables? I have binary variable in other constraints of my problem?
– KGM
Dec 10, 2020 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. Dec 10, 2020 at 13:52