Is it possible to express inequality constraints as equalities? I have a system of linear equations that I am trying to solve where the system is subject to a set of inequality constraints. The constraints are not to be placed on the estimated variables themselves but rather on the product between the variables and some minimum and maximum values in my dataset.
$$ x = min_x \frac{1}{2}||Dx-d||_2^2 $$ $$ s.t. $$ $$ Ax \leq c $$
I am hoping it is possible to somehow express these inequalities as equalities so that I can just include them in the original system of equations represented by D and d.
I don't know if this is what you're looking for, but here's an approach: if $A$ has $m$ rows, introduce a slack variables $s_1,\dots,s_m$. Rewrite the constraint as $Ax + \bar s = c$, where $$ \bar s = (s_1^2,s_2^2,\dots,s_m^2)^\top. $$