Linearizing $y=\sum_{i=1}^n(z+c)\left(\frac{r_i^2}{1-r_i}\right)\phi_i$

Question

Variables $0\le x< 1$, $y,z\ge 0$. We have a constraint $$y=(z+c)\frac{x^2}{1-x},$$ where constant $c>0$.

We partitioned $x$ into $n$ intervals of equal length and defined a new variable $\phi_i=1$ for $i=1,\ldots,n$ iff $x\in(r_i-1,r_i]$ (we set $x=r_i$) and zero otherwise.

So we have this constraint $$y=\sum_{i=1}^n(z+c)\left(\frac{r_i^2}{1-r_i}\right)\phi_i$$

How can we linearize this new constraint?

If we define a new non-negative variable $\xi_i=z\phi_i$ and add the required 3 additional constraints with $M$, can we rewrite the second constraint above as $$y=\sum_{i=1}^n\left(\frac{r_i^2}{1-r_i}\right)(\xi_i+c\phi_i)$$

Answer 1

You are on the right track. Linearizing the product of a continuous variable ($\xi_i$) and a binary variable ($\phi_i$) is a FAQ. See, for instance, How to linearize the product of a binary and a non-negative continuous variable?. It requires that you be able to slap upper and lower bounds on $\xi_i$.