Mixed Integer Programming with product of a binary variable and multiple continuous variables

Question

Suppose we have a binary variable $x$ and two non-negative continuous variables $y_1\ge 0$ and $y_2 \ge 0$. How can we linearize $xy_1 y_2$ ?

FYI, this is a follow up question to this: How to linearize the product of a binary and a non-negative continuous variable?

Answer 1

As Johan Löfberg said, it cannot be done directly. You can get an approximate solution in two steps.

1. First, approximate the product of $y_1$ and $y_2$ using a new variable $z.$ See, for instance, this question, and specifically the answers involving McCormick envelopes.
2. Now linearize the product of $x$ and $z$ using the link in your question.

Answer 2

I would start with a non-convex solver like Gurobi. Gurobi can only do quadratic terms, but that is not a real limitation:

 z1 = x*y1
 z2 = z1*y2