I have the following set of equations:
\begin{align*}
&\alpha_t \cdot (1-\theta) + \beta_t\cdot \theta = c_t \; \quad t = 1, \dots, T\\
&{\beta_t \cdot \theta^2 \over \alpha_t \cdot (1-\theta)^2 + \beta_t \cdot \theta^2} = k_t\; \quad t=1, \dots, T\\
&\sum_t^T \alpha_t = \sum_t^T \beta_t = 1\\
& 0 < \theta < 1
\end{align*}
where $c_t$ and $k_t$ are constants, $\sum_t c_t = 1$, and we want to solve for $\alpha_1, \dots, \alpha_T, \theta, \beta_1, \dots, \beta_T$. I want to solve the above equations using cvxpy but I'm not sure how to formulate the above into the form cvxpy wants.
Any suggestions will be greatly appreciated!
(Too long for a comment.) $\;$ Don't know that there is an advantage to formulate it as a multi-variate convex optimization problem, since it can be reduced to a single univariate rational equation.
Let $\,a_t = (1-\theta)\,\alpha_t\,$ and $\,b_t=\theta\,\beta_t\,$, then each pair of equations is a linear system in $\,a_t, b_t\,$:
$$ \begin{align} \begin{cases} a_t+b_t &= c_t \\ \theta\, b_t &= k_t\,\big((1-\theta)\,a_t+\theta\,b_t\big) \end{cases} \;\;\iff\;\; \begin{cases} a_t+b_t &= c_t \\ (1-k_t)\theta\, b_t &= k_t\,(1-\theta)\,a_t \end{cases} \end{align} $$
Eliminating $\,b_t\,$ between the equations:
$$ (1-k_t)\theta\,(c_t-a_t) = k_t\,(1-\theta)\,a_t \;\;\iff\;\; a_t = \frac{(1-k_t)c_t\theta}{(1-2k_t)\,\theta+k_t} $$
Substituting the above in $\,\sum_t \alpha_t = 1\,$ gives the equation in $\,\theta\,$:
$$ 1 = \sum_{t=1}^T \alpha_t = \sum_{t=1}^T \frac{a_t}{1-\theta} = \sum_{t=1}^T \frac{(1-k_t)c_t\theta}{(1-\theta)\big((1-2k_t)\,\theta+k_t\big)} = \frac{\theta}{1-\theta} \, \sum_{t=1}^T \frac{(1-k_t)c_t}{(1-2k_t)\,\theta+k_t} $$
Let $\,f(\theta)\,$ be the RHS of the latter, with $\,f(0)=0\,$. Under the additional assumption that $\,c_t, k_t \in (0,1)\,$ the limit $\,\lim_{\theta \to 1-} f(\theta)=+\infty\,$, so the equation $\,f(\theta)=1\,$ has a root $\,\theta \in (0,1)\,$.