For some positive definite matrix $A \in \mathbb{R}^{n \times n}$ I am solving the quadratic problem $$ \min_{\{x \in \mathbb{R}^n| Bx \le d\}} x^T A x + x^T b $$
Now, in addition to the linear constraints $$ Bx \le d$$ I have another constraint: Given some vector $y \in \mathbb{R}^n$ I need to make sure that the solution satisfies $$\sum_{i=1}^{n} |x_i - y_i| \le c$$ The solver I am using unfortunately only admits linear constraints.
I had the idea of rewriting $x = x^{+} - x^{-}$ where $(x^{+}, x^{-}) \in [0, \infty)^{2n}$ are both positive, but this makes things a bit messy. Is there an easy way to incorporate this constraint?
There is no "non-messy" way as you have to introduce more variables and constraints.
Either you use the strategy you talk about $$x-y=x^+-x^-, x^+\geq 0, x^-\geq 0, \sum_{i=1}^n x_i^++x_i^-\leq c$$
or the alternative epigraph approach $$-z \leq x-y\leq z, \sum_{i=1}^n z_i\leq c$$