# What is the difference between root relaxation and LP relaxation

(I apologize. I saw this question but, I do not know these may be the same or not.)

I am trying to solve a MIP problem and have an issue about that. The problem's LP relaxation has the objective value about $$63$$ but, when the problem is solved by MIP solvers like CPLEX, Gurobi and SCIP, the relax solution is reported $$59$$. (even though I disable pre-solving, cuts and heuristics).

Root relaxation: objective 5.900000e+01


In all of the cases, the value of the variables are the same and the problem is solved optimality.

I was wondering if, is there any way to get the root relaxation value equal to $$63$$ by using MIP solvers or this is an advanced technique in which the solvers use to speed up the solving time?

• How did you calculate the LP value? Gurobi has the command m.relax() when you optimize this you get 63? Jun 5 '20 at 21:53
• @user3680510, thanks. I'm using GAMS and the syntax for solving the LP relaxation is rmip. By using this you could get the value of the root node and it is $63$ but, by setting the syntax to mip the value is $59$. I think the value of the root node in LP relaxation or MIP directly should be the same but those are different!!! Jun 5 '20 at 22:01
• I would try to write an lp file and read in this file directly into the solver and see if this behaves the same way. Jun 5 '20 at 22:31
• @user3680510, I do that and the behaviour is the same!!! Jun 6 '20 at 3:51
• Please keep in mind that in a MIP, more preprocessing etc. can be done than in an LP. Thus, the MIP root node can be stronger than the LP relaxation.
– T_O
Jun 6 '20 at 9:45