@giuseppebilotta@fediscience.org this is linear programming. If you solve your initial linear problem (expressible in matrix form as Ax = b) as far as possible, you end up with a linear space of solutions with some dimension n (less than the number of variables you started with, but more than zero, because it's underdetermined). You want to consider only solutions x where you minimize some linear function of x, and also put linear conditions on x. This turns it into a classic linear programming problem. There are very effective software packages for solving linear programming problems, but you generally have to massage your problem into a standard form, which can take some doing if you're not familiar with this.