Any class of model, continuous or discrete, linear or quadratic, has the potential to result in a solution status of unbounded. An unbounded discrete model must have a continuous relaxation that is also unbounded. Therefore, the discussion here will assume that you will first relax any discrete elements, and thus you are dealing with an unbounded continuous optimization problem, when trying to diagnose the cause.

A declaration of unboundedness means that ILOG CPLEX has detected that the model has an unbounded ray. That is, given any feasible solution x with objective z, a multiple of the unbounded ray can be added to x to give a feasible solution with objective z-1 (or z+1 for maximization models). Thus, if a feasible solution exists, then the optimal objective is unbounded.

When a model is declared unbounded, ILOG CPLEX has not necessarily concluded that a feasible solution exists. Users can call methods or routines to discover whether ILOG CPLEX has also concluded that the model has a feasible solution.