I am working with the Bin Packing Problem, using a branch-and-price algorithm. There are instances that feasible tolerances of the Gurobi do not allow me solves completely the linear relaxation. That is, exists columns with reduced cost of absolute value less than 10^-9, and simplex does not pivot these columns.
Because of this, the root "optimal" solution can have the value X + c*10^-8, where X is the real optimal value and c is a constant greater than 1. This is expected since the dual solution is infeasible. There are ways to calculate a valid lower bound, such as the dual lagrangian lower bound or the dual safe bounds.
My doubt is, does the Gurobi MIP Solver uses safe bounds? If I pass the complete formulation to Gurobi, is guaranteed that it will not prune incorrectly a node in branch-and-bound tree because the simplex is unable to solve to optimality the relaxation?
Or does the Gurobi not guarantee correctness in instances with numerical issues?
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