# Does CVX allow Beta function?

N=4;
K=3;

\lambda=[100\hspace{3mm} 50 \hspace{3mm} 10];

\delta_k=2^{b_k}{\text{ Beta}(2^{b_k},\frac{1}{N-1})};

I_k=\lambda_k\delta_k;

My optimization problem is

\min_{b_k}\hspace{3mm}\max_{k\in\{1,2,\cdots,K\}}I_k

Here, b_k is the optimization variable.

How to express the objective function in CVX?

The list of supported functions is in the documentation, and to the best of our knowledge, it is complete. And do you know if your model is convex?

Although this looks hopeless (to me) for CVX, I think the log of the objective function is actually convex, but I have not definitively proven it.

Beta is a log-convex function of its first argument. We could take the log of δk, whose first term is convex, which leaves us the matter of dealing with the 2^b_k as the first argument of log of Beta in the 2nd term. I used MAPLE to symbolically compute the 2nd derivative of log(δk) using N=4,which produced an expression involving Digamma functions, and then numerically evaluated it at various argument values from 0 up to infinity (well not quite). Due to potential for numerical instability, I wound up doing the calculations in 2000 digit floating point, which I think wound up being reliable. I believe the 2nd derivative is positive at zero, monotonically (?) declining to zero as b_k goes to infinity. The 3rd derivative appears to always be negative, further supporting that the 2nd derivative is monotonically decreasing. Therefore, this appears to be convex, although I haven’t proven it. If it is convex, then the max over k will also be convex, and therefore the log of the objective would be convex. If this were my own problem, I’d put in the extra effort to increase confidence in that conjecture, and perhaps prove it analytically.

Ah, OK, well in that case, I’ll take off the “Nonconvex” specification, although I really I’d like it to be a catchall for “Nonconvex problems and other problems CVX is extremely unlikely to be able to solve, by design”

This certainly falls under the category of problems discussed in the FAQ: problems that cannot be expressed using the DCP rules, even though they are convex.