Minimally relaxing an infeasible linear program

Question

I am trying to debug a piece of code to 'relax' an infeasible linear program formulated in the standard formulation

Program 1 (that we assume infeasible):

$max/min$ $c^{T}r \\ s.t. Ar \leq, =, \geq b\\ l \leq r \leq u$

The following problem would find the minimum set of relaxation on upper and lower bounds and constraints of problem 1 to make it feasible.

Program 2

$min$ $c^T r + \lambda ||v||_0 + \alpha (||p||_0 + ||q||_0) \\ s.t. A r + v \leq, =, \geq b \\ l - p \leq r \leq u + q \\ v \in R^m \\ p,q \in R_+^n $

Now my question is:

Does the solution to problem 2 ensure solely a minimal set of bounds and constraints to relax? e.g. variable 2 should be relaxed.

Or

Does problem 2 find a set of bounds and constraints to relax and the minimal set of adjustements on the those bounds and constraints to find a feasible LP? e.g. variable 2 should be relaxed by a minimum of 2 units.

Answer 1

So the answer would be that Program 2 finds the minimal cardinality solution without necessarily ensuring a minimal adjustement to the bounds andconstraints.