# DCP representation of a convex quotient of affine functions

I am trying to represent the following inequality:

$$\frac{x}{1-x} \leq y \qquad\mathrm{with}\qquad 0

The function on the left is convex (its second derivative is always positive over the domain $$0), meaning that the described set is convex. (Without the domain constraint, however, the domain is far from convex…) Drawing it gives the same conclusion:

However, I cannot find a DCP representation of this set (even using nonstandard cones).

I can do it with a slight modification (replacing $$x$$ with $$x^2$$, two quad-over-lin constraints do the trick, CVX also accepts the constraint as $$y\geq\frac{x^2}{z}$$ and $$z\leq 1-x^2$$), but it is not describing the same set, and the difference is really significant in my case.

Clearly, the problem is equivalent to $$\begin{array}{rcl} \frac{1-t}{t} & \leq & y \\ 1-x & = & t \\ \end{array}$$ which in turn is equivalent to $$\begin{array}{rcl} \frac{1}{t} -1 & \leq & y \\ 1-x & = & t \\ \end{array}$$ This is clearly SOCP representable.

You should end up with $$\begin{array}{rcl} 2 (1-x) (1+y) & \geq & \sqrt{2}^2 \\ 1-x & \geq & 0 \\ 1+y & \geq & 0 \\ \end{array}$$ which is the same as saying $$\left [ \begin{array}{c} 1-x \\ 1+y \\ \sqrt{2} \\ \end{array} \right ] \in RQ$$ where RQ is a rotated quadratic cone as defined in the Mosek modelling cook book.

• Thanks for your detailed answer :)! – dourouc05 Apr 28 '20 at 17:49

You can model your constraint with the second order cone constraint $$\sqrt{2^2 + (x+y)^2} \le 2-x+y,$$ and a lower bound on $$x$$.

I found this by first multiplying both sides by $$1-x$$ to obtain $$(1-x)y - x >= 0$$. As we have a quadratic inequality with a convex feasible region, this hints to a quadratic or second order cone.

Rewriting this constraint gives $$(1-x)(1+y) \ge 1$$. Then I used Equation (7) in this paper to turn the constraint into a second order cone constraint.

• Thanks for your answer :)! – dourouc05 Apr 28 '20 at 17:49