An LP can have an unbounded feasible region but still have a unique, finite optimum because the objective function may improve only toward a bounded part of the feasible region. If the objective gradient points away from every unbounded ray, moving indefinitely along those rays does not improve the objective. Therefore, the objective reaches its best value at a specific boundary point, giving a finite and potentially unique optimal solution.

Even though the feasible region stretches infinitely in certain directions, the objective function gradient acts as a directional compass.

Because the gradient points away from every unbounded ray, moving outward along any infinite path continuously worsens the objective value. The optimization process is therefore pulled in the exact opposite direction—back toward the closed, bounded boundaries of the feasible region—where it is forced to stop at a single, finite vertex.