ILOG CPLEX 11.0 User's Manual

What Is Unboundedness?Any class of model, continuous or discrete, linear or quadratic, has the potential to result in asolution status of unbounded. An unbounded discrete model must have a continuousrelaxation that is also unbounded. Therefore, the discussion here will assume that you willfirst relax any discrete elements, and thus you are dealing with an unbounded continuousoptimization problem, when trying to diagnose the cause.Note: The reverse of that observation that an unbounded discrete model necessarily havingan unbounded continuous relaxation is not necessarily the case: a discrete optimizationmodel may have an unbounded continuous relaxation and yet have a bounded optimum.A declaration of unboundedness means that ILOG CPLEX has detected that the model hasan unbounded ray. That is, given any feasible solution x with objective z, a multiple of theunbounded ray can be added to x to give a feasible solution with objective z-1 (or z+1 formaximization models). Thus, if a feasible solution exists, then the optimal objective isunbounded.When a model is declared unbounded, ILOG CPLEX has not necessarily concluded that afeasible solution exists. Users can call methods or routines to discover whetherILOG CPLEX has also concluded that the model has a feasible solution.◆◆In Concert Technology, call one of these methods:●●●isDualFeasibleisPrimalFeasibletry/catch the exceptionIn the Callable Library, call the routine CPXsolninfo.Avoiding UnboundednessUnboundedness can be viewed as an under-constrained condition; such an outcome can befrom a modeler forgetting to include one or more constraints in the model. Thereforecarefully checking that your problem formulation is complete is a good first step indiagnosing unboundedness.The default variable type in CPLEX has a lower bound of 0 (zero) and an upper bound ofinfinity. If you declare a variable to be of type Free, its lower bound is negative infinityinstead of 0 (zero). A model can not be unbounded unless one or more of the variables haseither of these infinite bounds. Therefore, one straightforward tactic in avoidingunboundedness is to assign finite bounds to every variable in your model; if no variable cango on an unbounded ray to infinity, then your model can not be unbounded.388 ILOG CPLEX 11.0 — USER’ S MANUAL