You may wish to pose your question to another forum such as StackExchange: Mathematics or OR Exchange.

The purpose of this forum is to help people become *more effective users of CVX* and related tools. It is not a general forum about optimization, or even convex optimization, but specifically about the aforementioned software. For that reason, it is a rather firm prerequisite that when asking about a particular model, an attempt should already have been made to implement it in CVX (or TFOCS, for instance).

Unfortunately CVX is simply not a tool that can be used effectively without a good grasp of basic convex analysis and optimization. While we genuinely believe that CVX greatly simplifies the use of convex optimization in practice, it can be very frustrating to use without this knowledge.

Having said that, I do have a couple of observations about the problem that I will share.

First, it is obvious that your model *is not feasible*. For instance, R=(P_sP_r)/(P_sP_r+P_s+P_r+1) clearly satisfies 0\leq R<1, because the numerator is strictly less than the denominator, and both are nonnegative. Therefore, it is not possible for 1/R\leq 1/10, or equivalently, R\geq 10.

Second, suppose the model were corrected to remedy that problem. The question remains how to implement the 1/R quantity in CVX:

$$1/R=(P_sP_r+P_s+P_r+1)/(P_sP_r) =1+P_s^{-1}+P_r^{-1}+P_s^{-1}P_r^{-1}$$

The individual terms P_s^{-1} and P_r^{-1} are handled using the `inv_pos`

function. Because `inv_pos`

implicitly constrains its argument to be positive, there is no need to add constraints like `Ps >= 0`

to your CVX model; they would be redundant.

What about P_s^{-1}P_r^{-1}? There are actually two ways to implement this currently. The first takes advantage of the fact that

$$P_s^{-1}P_r^{-1}=\left(\det\begin{bmatrix} P_s & 0 \ 0 & P_r \end{bmatrix}\right)^{-1}.$$

Given this, we can write it as `det_inv([Ps,0;0,Pr])`

. Alternatively, we can write it as a *nonlinear composition*

$$P_s^{-1}P_r^{-1}=(\sqrt{P_sP_r})^{-2}$$

which becomes `pow_p(geo_mean([Ps,Pr]),-2)`

in CVX. This obeys the nonlinear composition rules in the DCP ruleset: `pow_p(.,-2)`

is convex and nonincreasing, and `geo_mean`

is concave, therefore they can be combined.

Combining these, we can write

```
Rinv = 1 + inv_pos(Ps) + inv_pos(Pr) + pow_p(geo_mean([Ps,Pr]),-2)
```

in CVX. This is a convex expression that can be used anywhere convex expressions are allowed according to the DCP ruleset.