Suggestion for choosing (building) optimization function

Question

I would like to build a supervised learning model M satisfying the following conditions:

1. Training data $\{X, Y\}$, where $x \in R^m$ and $y \in R^n$

2. Assume: $M(x) = p$, then: $0 < p[k] <= y[k]$, for all $k = 1,\dots, n$. Here I mean $p[k]$ as close as possible to $y[k]$

Could you please suggest what are the "best" cost function and optimization method that I can use to train this model $M$?

Thank you.