Suppose I have a convex hull constraint, that requires the complex number variable is located in the regular polygon on complex plane with K vertices and the first vertex will be at real axis. Then how we formulate it both in math and for the CVX solver?
Maybe we can take the itersection of the halfspaces from each side of the polygon? But it seems to be a little bit complicated, especially when K grows large. Is there a neat way to formulate it?
As K will not be too large, right now I think maybe using the intersection of halfplanes is fine.
Perhaps you can help clarify what your problem is by explicitly showing the mathematical formulation of a convex optimization problem which does what you want, even if that formulation is not “neat”. And also show us your CVX code for that formulation, even if it is not “neat”., Then perhaps a reader can weigh in as to whether they know of a “neater” formulation.
\Is what you want just a set of inequality constraints?
Yes, to express this convex polygon, right now I am thinking using a bunch of inequalities to do that, which can be formulated as
\mathcal{F}_{\mathcal{V}_k}=\left\{v\left|\operatorname{Re}\left(t_k^* v\right) \leq \cos \left(\frac{\pi}{K}\right), t_k=e^{j \frac{\left(\theta_k+\theta_{k+1}\right)}{2}},\right| \mid v \mid \leq 1\right\}, \ k=1, \ldots, K.
where the radius is 1, v is the complex variable, K is the number of the vertices, the first vectex is at positive real axis. \theta_k represents the phase of k-th vertex on complex plane, and the t_k is the outer normal of the connectivity between \theta_k and \theta_{k+1}.
As v is the entry of a vector variable, which means I have to perform all the inequalities above for each entry. So that’s why I wondered whether there will be a “neater” formulation here.
Horizontally concatenate the t_k and RHS and use a single (element-wise, which is the default) matrix inequality.
Thanks Mark, I will try it!
I.e, all the t_k is a single matrix variable.
Yep, then a single inequality \text{Re}(\textbf{t}v)\le \textbf{z} can express the inequality above, where \textbf{t}=[t_1^{*},\ldots,t_K^{*}] and \textbf{z} = [\cos(\frac{\pi}{K}),\ldots,\cos(\frac{\pi}{K})].