Handling non-convex inequality constraint in otherwise convex problem

Question

I have an inequality constrained optimization problem, which I believe to be convex except for a single constraint I am having trouble handling. The objective function looks like this:

$\sum_{i=1}^{n} \frac{c_i}{x_i + y_i}$

Where $x,y \geq 0$ are the optimization variables and $c_i \geq 0$ are constants. For brevity, I'm going to omit some of the other constraints in the problem, which are mostly lower and upper bounds on the variables along with a single convex inequality constraint. However, there is one final set of constraints which I'm not sure how to handle, given by:

$\left(\frac{y_i – 1}{x_i + y_i – 1}\right) a_i – b_i \leq 0$

Here $a_i $ and $b_i$ are positive constants and my other constraints enforce that the fraction is also positive. From what I have read, this looks like a linear fractional function, which is quasilinear. I'm an optimization novice and have had difficulty finding references for handling these kinds of functions in constraints. What I have thought about so far:

• I found some resources for these functions when they are the objective of the problem, but I don't see how those techniques are applicable here. Maybe there is a change of variable I could consider?
• I could consider some kind of affine approximation of this function, which would make the overall problem convex. I would then need to quantify the worst case distance between the approximation and the actual function.
• I could pretty easily relax the fraction to be $\frac{y_i}{x_i + y_i}$ but it's not immediately obvious how that might simplify things.

Looking for guidance or further references for handing this situation.

Answer 1

You asked me to formalize my comment as an answer: Your constraints are (for some subset of indices $i$): $$ \frac{(y_i-1)a_i}{x_i+y_i-1} \leq b_i$$ There is a potential singularity if the denominator of the fraction is zero. However, if we assume that other constraints ensure that $x_i+y_i-1>0$ always, then we can multiply the inequality by the positive number $x_i+y_i-1$ to equivalently get $$ (y_i-1) a_i \leq b_i(x_i+y_i-1)$$ which is a linear (hence convex) inequality constraint.