How do handle maximum in a quadratic programming problem by adding auxiliary variables?

Question

I am solving a quadratic programming problem of the form $$\min_x ||Rx - b||^2$$ subject to a couple of constraints which I've already defined. I'd like to change my objective function $||Rx-b||^2$ to $||\max(Rx - b, 0)||^2$ where the max is taken componentwise over the vector $Rx - b$. To give an example of what I mean, we might have $\max( (2, -3, 1), 0) = (2, 0, 1)$.

I'm following a paper where the author has done this, and the author says that you can add "auxiliary variables in the canonicalisation process" to do this, but I don't see how to proceed and keep this within the framework of convex quadratic optimisation.

What can I do? What might the author have meant by adding additional auxiliary variables in the canonicalisation process to achieve this?

Answer 1

Add variables $y_i \ge 0$ and constraints $$y_i \ge (Rx - b)_i \quad \forall i,$$ and change the objective to minimize $\sum_i y_i^2.$