In an optimization problem were the objective function is to be maximized the optimal value is the least upper bound of the objective function values over the entire feasible region. If there is no upper bound, then we say that the optimal value is +∞, while if the feasible region is the empty set, we define the optimal value of a maximization problem to be −∞. Conversely, in an optimization problem were the objective function is to be minimized the optimal value is the greatest lower bound of the objective function values over the entire feasible region. If there is no lower bound, then we say that the optimal value is −∞, while if the feasible region is the empty set, we define the optimal value of a minimization problem to be +∞.

Therefore, every optimization problem has a well-defined optimal value. But not every optimization problem has an optimal solution. For example, consider the optimization problem min {e x : x ∈ R}. this problem has an optimal value of zero, but there is no optimal solution.