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I haven't been able yet to represent those using DCP, but I found some interesting things:

This is the tricky one. First note that:

\begin{equation} \log(t)\log(1-t) = \frac{1}{2}\left(\log^2(t(1-t)) - \log^2\left(\frac{t}{1-t}\right)\right), \end{equation}

since $2cd = (c+d)^2 - (c-d)^2$, for any $c,d$.

Now form the optimization problem: \begin{align} \min_{z}\,-\frac{z}{2} &\qquad\text{s.t.: }\\ &2z \leq w-v,\\ &\,\sqrt{v} \geq \left\lvert\log\left(\frac{t}{1-t}\right)\right\rvert,\tag{1}\\ &\,\sqrt{w} \leq \lvert\log(t(1-t))\rvert\tag{2}, \end{align}

Now we deal with constraints (1) and (2):

For (1) Note that: \begin{equation} \lvert\log(t)-\log(1-t)\rvert = \text{ptp}{([-\log(t),-\log(1-t)])}, \end{equation} where $\text{ptp}$ is the peak-to-peak atom in CVX, which is convex. Since $-\log(\cdot)$ is a convex atom, we're set. I can't seem to find cp.ptp when I try to use it in CVXPY, though. Is it really implemented?

Now for (2), note that, since $t \in (0,1)$, we have that : \begin{equation} \lvert\log(t(1-t))\rvert = -\log(t(1-t)). \end{equation}

Thus, we can write (2) as:

\begin{align} &\sqrt{w} + \log(t(1-t)) &\leq 0 &\iff\\ \sqrt{w}t + \sqrt{w}(1-t) &+\log(t) +\log(1-t) &\leq0&\iff\\ \frac{t}{t}\log\left(\frac{t}{e^{-\sqrt{w}t}}\right)&+ \frac{1-t}{1-t}\log\left(\frac{1-t}{e^{-\sqrt{w}(1-t)}}\right)&\leq0 &\iff\\ \frac{1}{t}D_\text{KL}(t \lvert\rvert e^{t h(w)}) &+ \frac{1}{1-t}D_\text{KL}((1-t) \lvert\rvert e^{(1-t)h(w)})&\leq0 &\iff\\ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{t}} &+ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{1-t}}&\leq0, \end{align}

where $h(w) = -\sqrt{w}$, $D_\text{KL}(x \lvert\rvert y)$ is the relative entropy between x and y, $g(x,y) = D_\text{KL}(x \lvert\rvert e^y)$, and $\mathcal{P}$ is the perspective transform of a function.

I think this might work, it's all the composition of simpler convex functions.

I found some interesting things:

This is the tricky one. I had given up already, when I found this wikipedia page on the Dilogarithm function.

\begin{equation} \text{Li}_2(z) = -\int_{0}^{z}\frac{\log(1-t)}{t}dt = -\int_{0}^{1}\frac{\log(1-z t)}{t}dt, \end{equation}

and:

\begin{equation} \text{Li}_2(z) + \text{Li}_2(1-z) = \frac{1}{6}\pi^2 - \log(z)\log(1-z) \end{equation}

Now we can approximate the integral with numerical integration (trapezoidal rule): \begin{equation} -\int_{0}^{z}\frac{\log(1-t)}{t}dt \approx -\frac{1}{N_t}\sum_{k=1}^{N_t}\frac{1}{2}\left(\frac{\log(1-z t_k)}{t_k} + \frac{\log(1-z t_{k-1})}{t_{k-1}}\right), \end{equation} where $t_k$ for $k\in\{0,N_t\}$ discretizes the interval $[0,1]$.

Now this approximation is a convex DCP expression.

I guess the only thing missing now is to write the remaining constraint for the change of variable.

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?
3. If it can't be put into a DCP compliant formulation, what would you recommend that I do to solve this problem numerically (especially question 2)?

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 right-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 $T$ and adding a (quasilinear) constraint to account for the change of variable. Question 2 finally reduces to finding a DCP compliant representation of $T$.

Update: Progress on trying to find a DCP representation of $\log^2(t)$ and $-\log(t)\log(1-t)$:

I haven't been able yet to represent those using DCP, but I found some interesting things:

a. $\log^2(t)$:

We can write this as: \begin{equation} \min_{z} z \qquad\text{s.t.: } w\geq -\log(t),\,w\geq 0,\,z\geq w^2, \end{equation} for fixed $t \in (0,1)$.

b. $-\log(t)\log(1-t)$:

This is the tricky one. First note that:

\begin{equation} \log(t)\log(1-t) = \frac{1}{2}\left(\log^2(t(1-t)) - \log^2\left(\frac{t}{1-t}\right)\right), \end{equation}

since $2cd = (c+d)^2 - (c-d)^2$, for any $c,d$.

Now form the optimization problem: \begin{align} \min_{z}\,-z &\qquad\text{s.t.: }\\ &2z \leq w-v,\\ &\,\sqrt{v} \geq \left\lvert\log\left(\frac{t}{1-t}\right)\right\rvert,\tag{1}\\ &\,\sqrt{w} \leq \lvert\log(t(1-t))\rvert\tag{2}, \end{align}

Now we deal with constraints (1) and (2):

For (1) Note that: \begin{equation} \lvert\log(t)-\log(1-t)\rvert = \text{ptp}{([-\log(t),-\log(1-t)])}, \end{equation} where $\text{ptp}$ is the peak-to-peak atom in CVX, which is convex. Since $-\log(\cdot)$ is a convex atom, we're set. I can't seem to find cp.ptp when I try to use it in CVXPY, though. Is it really implemented?

Now for (2), note that, since $t \in (0,1)$, we have that : \begin{equation} \lvert\log(t(1-t))\rvert = -\log(t(1-t)). \end{equation}

Thus, we can write (2) as:

\begin{align} &\sqrt{w} + \log(t(1-t)) &\leq 0 &\iff\\ \sqrt{w}t + \sqrt{w}(1-t) &+\log(t) +\log(1-t) &\leq0&\iff\\ \frac{t}{t}\log\left(\frac{t}{e^{-\sqrt{w}t}}\right)&+ \frac{1-t}{1-t}\log\left(\frac{1-t}{e^{-\sqrt{w}(1-t)}}\right)&\leq0 &\iff\\ \frac{1}{t}D_\text{KL}(t \lvert\rvert e^{t h(w)}) &+ \frac{1}{1-t}D_\text{KL}((1-t) \lvert\rvert e^{(1-t)h(w)})&\leq0 &\iff\\ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{t}} &+ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{1-t}}&\leq0, \end{align}

where $h(w) = -\sqrt{w}$, $D_\text{KL}(x \lvert\rvert y)$ is the relative entropy between x and y, $g(x,y) = D_\text{KL}(x \lvert\rvert e^y)$, and $\mathcal{P}$ is the perspective transform of a function.

I think this might work, it's all the composition of simpler convex functions.

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?
3. If it can't be put into a DCP compliant formulation, what would you recommend that I do to solve this problem numerically (especially question 2)?

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 right-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 $T$ and adding a (quasilinear) constraint to account for the change of variable. Question 2 finally reduces to finding a DCP compliant representation of $T$.

Update: Progress on trying to find a DCP representation of $\log^2(t)$ and $-\log(t)\log(1-t)$:

I haven't been able yet to represent those using DCP, but I found some interesting things:

a. $\log^2(t)$:

We can write this as: \begin{equation} \min_{z} z \qquad\text{s.t.: } w\geq -\log(t),\,w\geq 0,\,z\geq w^2, \end{equation} for fixed $t \in (0,1)$.

b. $-\log(t)\log(1-t)$:

This is the tricky one. First note that:

\begin{equation} \log(t)\log(1-t) = \frac{1}{2}\left(\log^2(t(1-t)) - \log^2\left(\frac{t}{1-t}\right)\right), \end{equation}

since $2cd = (c+d)^2 - (c-d)^2$, for any $c,d$.

Now form the optimization problem: \begin{align} \min_{z}\,-\frac{z}{2} &\qquad\text{s.t.: }\\ &2z \leq w-v,\\ &\,\sqrt{v} \geq \left\lvert\log\left(\frac{t}{1-t}\right)\right\rvert,\tag{1}\\ &\,\sqrt{w} \leq \lvert\log(t(1-t))\rvert\tag{2}, \end{align}

Now we deal with constraints (1) and (2):

For (1) Note that: \begin{equation} \lvert\log(t)-\log(1-t)\rvert = \text{ptp}{([-\log(t),-\log(1-t)])}, \end{equation} where $\text{ptp}$ is the peak-to-peak atom in CVX, which is convex. Since $-\log(\cdot)$ is a convex atom, we're set. I can't seem to find cp.ptp when I try to use it in CVXPY, though. Is it really implemented?

Now for (2), note that, since $t \in (0,1)$, we have that : \begin{equation} \lvert\log(t(1-t))\rvert = -\log(t(1-t)). \end{equation}

Thus, we can write (2) as:

\begin{align} &\sqrt{w} + \log(t(1-t)) &\leq 0 &\iff\\ \sqrt{w}t + \sqrt{w}(1-t) &+\log(t) +\log(1-t) &\leq0&\iff\\ \frac{t}{t}\log\left(\frac{t}{e^{-\sqrt{w}t}}\right)&+ \frac{1-t}{1-t}\log\left(\frac{1-t}{e^{-\sqrt{w}(1-t)}}\right)&\leq0 &\iff\\ \frac{1}{t}D_\text{KL}(t \lvert\rvert e^{t h(w)}) &+ \frac{1}{1-t}D_\text{KL}((1-t) \lvert\rvert e^{(1-t)h(w)})&\leq0 &\iff\\ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{t}} &+ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{1-t}}&\leq0, \end{align}

where $h(w) = -\sqrt{w}$, $D_\text{KL}(x \lvert\rvert y)$ is the relative entropy between x and y, $g(x,y) = D_\text{KL}(x \lvert\rvert e^y)$, and $\mathcal{P}$ is the perspective transform of a function.

I think this might work, it's all the composition of simpler convex functions.

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?
3. If it can't be put into a DCP compliant formulation, what would you recommend that I do to solve this problem numerically (especially question 2)?

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 right-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 $T$.

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?
3. If it can't be put into a DCP compliant formulation, what would you recommend that I do to solve this problem numerically (especially question 2)?

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 right-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 $T$ and adding a (quasilinear) constraint to account for the change of variable. Question 2 finally reduces to finding a DCP compliant representation of $T$.

Update: Progress on trying to find a DCP representation of $\log^2(t)$ and $-\log(t)\log(1-t)$:

I haven't been able yet to represent those using DCP, but I found some interesting things:

a. $\log^2(t)$:

We can write this as: \begin{equation} \min_{z} z \qquad\text{s.t.: } w\geq -\log(t),\,w\geq 0,\,z\geq w^2, \end{equation} for fixed $t \in (0,1)$.

b. $-\log(t)\log(1-t)$:

This is the tricky one. First note that:

\begin{equation} \log(t)\log(1-t) = \frac{1}{2}\left(\log^2(t(1-t)) - \log^2\left(\frac{t}{1-t}\right)\right), \end{equation}

since $2cd = (c+d)^2 - (c-d)^2$, for any $c,d$.

Now form the optimization problem: \begin{align} \min_{z}\,-z &\qquad\text{s.t.: }\\ &2z \leq w-v,\\ &\,\sqrt{v} \geq \left\lvert\log\left(\frac{t}{1-t}\right)\right\rvert,\tag{1}\\ &\,\sqrt{w} \leq \lvert\log(t(1-t))\rvert\tag{2}, \end{align}

Now we deal with constraints (1) and (2):

For (1) Note that: \begin{equation} \lvert\log(t)-\log(1-t)\rvert = \text{ptp}{([-\log(t),-\log(1-t)])}, \end{equation} where $\text{ptp}$ is the peak-to-peak atom in CVX, which is convex. Since $-\log(\cdot)$ is a convex atom, we're set. I can't seem to find cp.ptp when I try to use it in CVXPY, though. Is it really implemented?

Now for (2), note that, since $t \in (0,1)$, we have that : \begin{equation} \lvert\log(t(1-t))\rvert = -\log(t(1-t)). \end{equation}

Thus, we can write (2) as:

\begin{align} &\sqrt{w} + \log(t(1-t)) &\leq 0 &\iff\\ \sqrt{w}t + \sqrt{w}(1-t) &+\log(t) +\log(1-t) &\leq0&\iff\\ \frac{t}{t}\log\left(\frac{t}{e^{-\sqrt{w}t}}\right)&+ \frac{1-t}{1-t}\log\left(\frac{1-t}{e^{-\sqrt{w}(1-t)}}\right)&\leq0 &\iff\\ \frac{1}{t}D_\text{KL}(t \lvert\rvert e^{t h(w)}) &+ \frac{1}{1-t}D_\text{KL}((1-t) \lvert\rvert e^{(1-t)h(w)})&\leq0 &\iff\\ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{t}} &+ \mathcal{P}\left(g(x,y),s\right)\bigg\rvert_{x=1, y=h(w), s=\frac{1}{1-t}}&\leq0, \end{align}

where $h(w) = -\sqrt{w}$, $D_\text{KL}(x \lvert\rvert y)$ is the relative entropy between x and y, $g(x,y) = D_\text{KL}(x \lvert\rvert e^y)$, and $\mathcal{P}$ is the perspective transform of a function.

I think this might work, it's all the composition of simpler convex functions.