Mathematical Programming

, Volume 99, Issue 2, pp 283-296

First online:

Safe bounds in linear and mixed-integer linear programming

  • Arnold NeumaierAffiliated withInstitut für Mathematik, Universität Wien Email author 
  • , Oleg ShcherbinaAffiliated withInstitut für Mathematik, Universität Wien

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Current mixed-integer linear programming solvers are based on linear programming routines that use floating-point arithmetic. Occasionally, this leads to wrong solutions, even for problems where all coefficients and all solution components are small integers. An example is given where many state-of-the-art MILP solvers fail. It is then shown how, using directed rounding and interval arithmetic, cheap pre- and postprocessing of the linear programs arising in a branch-and-cut framework can guarantee that no solution is lost, at least for mixed-integer programs in which all variables can be bounded rigorously by bounds of reasonable size.


linear programming mixed-integer programming rounding errors directed rounding interval arithmetic branch-and-cut lower bounds mixed-integer rounding generalized Gomory cut safe cuts safe presolve certificate of infeasibility