How can I transform my objective function to a convex optimization problem

Nonconvex
1

I have the following optimization problem

maximize_{\mathbf{w}_1,\mathbf{w}_2} \quad \min (\frac{|{\mathbf{v}^H \mathbf{w}_1}|^2}{|{(\mathbf{u}+\mathbf{v})^H \mathbf{w}_2}|^2+\sigma_1^2} , \frac{|{(\mathbf{u}+\mathbf{v})^H \mathbf{w}_2}|^2}{|{\mathbf{v}^H \mathbf{w}_1}|^2+\sigma_2^2})\\ s.t. \quad \| \mathbf{w}_1\|^2 +\| \mathbf{w}_2\|^2 \le P

where \mathbf{u,v} are known vectors.

I was wondering if there is any way that I can transform the objective function into a convex function so that I can use CVX to solve my problem.

2

The two arguments of min don’t look concave to me, which hey would need to be for this to be a convex optimization problem. Of course you’re welcome to change the objective to something which is not mathematically equivalent, but which could be input to CVX. But it is not the place of the readers of this forum to judge which changes would be acceptable for your purposes.

1 Like
3

It is not convex. But it looks odd to me also because when you reach the maximum, those two expressions inside of min might should be equal, otherwise you can always increase the min by adjusting the smaller expression, by changing its w1 and w2 separately but with a different direction.

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4

Thanks.

Actually, I was thinking about this, and I believe you may be right.

The solution must be where

\frac{|{\mathbf{v}^H \mathbf{w}_1}|^2}{|{(\mathbf{u}+\mathbf{v})^H \mathbf{w}_2}|^2+\sigma_1^2} =\frac{|{(\mathbf{u}+\mathbf{v})^H \mathbf{w}_2}|^2}{|{\mathbf{v}^H \mathbf{w}_1}|^2+\sigma_2^2}