Consider the following convex optimization problem in vectors $x$ and $y$
$$\begin{array}{ll} \text{minimize} & f(x,y)\\ \text{subject to} & g_1 (x)\le 0\\ & g_2 (y)\le 0\end{array}$$
where function $f$ is convex and functions $g_1$ and $g_2$ are linear. All three functions are continuous and differentiable.
Noting that the constraints are separate, if the dimensions of $x$ and $y$ are very large (e.g., more than thousand), then how can I handle this type convex optimization problem in a more efficient way?