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
f = 2abs(w’ * w-1) + 3(w’ * w-1);
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