I have the following objective that I want to maximize:

\begin{equation} \max_{U_T\in \mathbb{R}, x\in\mathbb{R}^T} J_\alpha(U_T) = \frac{\alpha}{\alpha-1}\log\left(\frac{\cosh(U_T)}{\cosh(\alpha U_T)^\frac{1}{\alpha}}\right) \,,\qquad \text{s.t:} \qquad U_T= Ax+b. \end{equation} where $A \in \mathbb{R}^{1\times T}$, $b\in\mathbb{R}$ and $\alpha>1$.

It is easy to show that this is, in fact, a pseudo-concave function by checking that: \begin{equation} \eta'_\alpha(u)(v-u)\leq0 \implies \eta_\alpha(v)\leq \eta_\alpha(u), \end{equation} where $\eta_\alpha(u) = \frac{\cosh(u)}{\cosh(\alpha u)^\frac{1}{\alpha}}$.

This, of course, implies the quasiconcavity of $J_\alpha(U_T)$.

Here's a plot of the function that I made with Wolfram Mathematica: $J_\alpha(x)$ for $\alpha=2$." />

-My questions are:

1. Is there a way to write this objective function as a DQCP compliant program in CVXPY?
2. When $\alpha \to 1^+$, $J_1(u) := -\log[\frac{e^{u\tanh(u)}}{\cosh(u)} ]$ pointwise. Is there a way to write that as a DQCP compliant program in CVXPY as well?

I'm using CVXPY since I'll add more complicated constraints later, but maybe the problem has a simple analytical solution as well.

-What I have so far:

I am especially interested in question 2, so I'm trying to solve that first.

Using the definitions of $\cosh$, write: \begin{align} J_1(u) &:= \lim_{\alpha\to 1^+}J_\alpha(u)\\ &= \log(\cosh(u)) -u \tanh(u)\\ &= \log\left(e^u + e^{-u}\right)-u \tanh(u) + K\\ &= -u\left(\log\left(\frac{1}{1 + e^{-2u}}\right)+ \tanh(u)\right) + K\\ &= -\frac{1}{2}f^{-1}(f(2u))(\log(f(2u))- \tanh(u)) + K, \end{align} where $K$ is a constant and $f(y) := 1/(1+e^{-y})$ is the logistic function, with inverse given by $f^{-1}(y) = \log(\frac{y}{1-y})$. By the well-known relation (Wiki): \begin{equation}\tanh(y) = 2f(2y) - 1,\end{equation} and we can conclude that: \begin{equation} J_1(u) = T(f(2u)) + K, \end{equation} where $T:[0,1]\to\mathbb{R}$ is: \begin{equation}T(z) = -\frac{1}{2}\log\left(\frac{z}{1-z}\right)\left(\log(z) - 2z-1\right).\end{equation}

T is in fact concave, as we can see from the graph of its second derivative in Mathematica:

$T(z)$" />

By further manipulation of $T$, we get the expression:

\begin{align} -2T(z) &= \log\left(\frac{z}{1-z}\right)(z + z - 1 + \log(z))\\ &= z\log\left(\frac{z}{1-z}\right) + (1-z)\log\left(\frac{1-z}{z}\right) + \log\left(\frac{z}{1-z}\right)\log(z)\\ &= 2\text{JS}(t \vert\vert 1-t) + \log^2(z) - \log(z)\log(1-z), \end{align} where $\text{JS}$ is the Jensen-Shannon entropy (simmetrized KL divergence).

Note the left-hand side is a sum of convex functions, as $\log^2(z)$ and $-\log(z)\log(1-z)$ are convex in $(0,1)).

Therefore, I think I would be able to write the original program by optimizing for T1 and adding a (quasilinear) constraint to account for the change of variable. Question 2 finally reduces to finding a DCP compliant representation of $T$.