Consider the optimization problem:
\begin{equation} \begin{array}{l} \min_{(x,y)\in \mathbb{R}^2_{+}} \quad x_{1}a_{1} + x_2a_{2} \\ \text {subject to } \quad\; y_{2} \ge \frac{1}{{x}_2}, \\ \quad \quad \quad\quad \quad y_{1} \ge y_{2} + \frac{1}{{x}_1}, \\ \quad \quad \quad\quad \quad y_0 \ge y_{1}, \\ \quad \quad \quad\quad \quad a_0 \ge y_0, \end{array} \end{equation} where $x,y$ are positive variables and $a$ are non-negative constants.
Is it possible to solve the problem in a distributed manner through e.g., the ADMM algorithm? (note a distributed solution is useful since the original problem is high dimensional, has many more inequality constraints with the same structure, and includes additional separable equality constraints).
What I have tried: We can reformulate the problem with auxiliary variables $z_1 = y_1$ and $z_2 = y_2$, so that: \begin{equation} \begin{array}{l} \min_{x,y} \quad \quad \;\;x_{1}a_{1} + x_2a_{2} \\ \text {subject to } \quad\; y_{2} \ge \frac{1}{{x}_2}, \\ \quad \quad \quad\quad \quad y_{1} \ge z_{2} + \frac{1}{{x}_1}, \\ \quad \quad \quad\quad \quad y_0 \ge z_{1}, \\ \quad \quad \quad\quad \quad a_0 \ge y_0, \\ \quad \quad \quad\quad \quad y_2 = z_2, \\ \quad \quad \quad\quad \quad y_1 = z_1, \end{array} \end{equation}
Now the constraints are "separated" and we can have three sets of variables $(x_2,y_2)$, $(x_1,y_1,z_2)$, and $(y_0,z_1)$, with three corresponding sub-problems. Does it make sense to proceed this way?
I don't really know how to handle inequality constraints since it seems that ADMM is mainly tailored to affine equality constraints. Adding slack variables for any constraint doesn't really work since the inequality constraints are non-affine. This paper generalizes ADMM to non-linear inequality-constrained problems but it seems to complicate the derivation and potentially the solution to the local sub-problems.
Maybe ADMM is not really suited to solve this kind of problem. Any idea on better distributed methods better suited to solve it? I thought of Generalized Benders decomposition but I am not sure how to formulate the problem.
