How to formulate an ADMM optimization problem involving inequality constraints?

Question

I have an optimization problem as shown below:

$\begin{alignat}{2} &\!\min_{x} &\qquad & J(x) \\ &\text{subject to} & & t_{1} = x_{11}C_{11}+ x_{12}C_{21},\\ & & & t_{1} > 0,\\ & & & t_{2} = x_{21}C_{21}+ x_{22}C_{22},\\ & & & t_{2} > 0\\ \end{alignat}$

Here, $J(x)$ is a convex function depending on the set of variables $x = \lbrace x_{11},~x_{12},~x_{21},~x_{22} \rbrace$. The entities $C_{11},~C_{12},~C_{21}$ and $ C_{22}$ are the constants belonging to the set of Real numbers. I want to solve the above optimization problem using the ADMM approach. But, the literature that I have referred to, has not given a clear method to reformulate an optimization problem shown above to the ADMM compatible form.

However, the slide 10 in the presentation here, has formulated an optimization problem with inequality constraints in the Dual decomposition form, but I require the ADMM formulation of the similar problem involving inequalities. Also, they are $\leq$ inequalities, while the optimization problem stated above has strict $>$ inequalities. Hence, I request an approach or some good literature from the community to deal with this kind of inequalities while formulating ADMM optimization problem.

Please note that I am a beginner in this field, and have a very little knowledge about it. So an approach with detailed explanation will be highly appreciated.

Answer 1

After reading the paper Distributed Convex Optimization with Many Non-Linear Constraints, I found the answer to my question. In the section 6, the authors clearly explains the approach they took to transform an optimization problem involving inequalities into the ADMM form. I would strongly recommend reading the whole paper to get a better understanding of the concept.

Answer 2

You can reformulate any > inequality as a < inequality by reversing the left-hand and right-hand sides. For instance, $t_1 > 0$ is equivalent to $-t_1 < 0$.

Note that practical solvers generally don't make a distinction between < and $\le$. So the solver will solve something like $-t_1 \le 0$. If you need $t_1$ to be strictly greater than 0, then you will need to pick some suitable small number, large enough to absorb solver tolerance in satisfying the inequality, and specify $-t_1 \le$ small_number