How to transform this convex objective function into cvx sentence?

a\left|\left\| w \right\|_2^2 - 1\right| +b \left(\left\| w \right\|_2^2 - 1 \right), where w is a complex vector.
I try to translate this one into cvx directly. However, it returns me ‘‘abs( convex )’’. Can some one help me?

variable w(10) complex
expression f
f = 2abs(w’ * w-1) + 3(w’ * w-1);
minimize f
subject to
norm(w) <= 10;

The first term of the objective function is not convex. Just to make this readily apparent, consider it as a function of a scalar variable x (in place of w). Then the first term has a W shape when plotted as a function of x (the bottoms of the W are at x = -1 and x = 1), which is not convex.

If a >= 0, b >= 0, b >= a, as is the case in your example, then I believe the overall objective function is convex, and is equal to

(b-a)*(w'*w-1) if norm(w) < 1
(a+b)*(w'*w-1) if norm(w) >= 1

but I don’t know how to express that in CVX.

The condition you said is satisfied. a\ge 0, b \ge 0, b > a. The whole term is convex. But I want to solve it in cvx, that is probably the major challenge.

What about (b-a)*(sum_square(w)-1) + 2*a*max(sum_square(w)-1,0)? Or perhaps more efficiently:

z = sum_square(w) - 1;
f = (b-a) * z + 2 * a * max(z, 0);

I should have thought of that. Although I think you need sum_square_abs because w is complex.

Thank you very much!!

let Q=ww’, then w’w=trace(Q) and ( w’ * w-1)=trace(Q)-1. With the assumption that trace(Q)>1. Your problem can be reformulated as follows:


variable Q(10,10) complex hermitian semidefinite
expression f
f = 2* (trace(Q)-1)+ 3*( trace(Q)-1);
minimize f
subject to
real(trace(Q))<= 10^.5


then you can calculate w from Q by using svd rank reduction

Actually, I don’t have that assumption.

Tray to relax your objective function by minimizing the upper bound in stead of the exact value. Your solution will be close to the optimal solution depending on the sharpness of your upper bound.

We actually have an exact answer already above.

‘’ I have to agree with Mark here. Reforumulating nonconvex problems is simply outside of the scope of this forum. For one thing, it’s extremely unlikely that it’s even possible ‘’

I’m lost. We have already solved it…

@mahmood964 , is there a reason you have just re-used mcg’s answer to your question in Reformulation of a non-convex constraint so that it become convex ? Has your account been taken over by a bot?

It might be. Please ignore that response.