This property is at the basis of the simplex method, which in fact works in the following way. First it looks for an initial vertex of P. Then, at each iteration, given a current vertex, tests if it satisfies some optimality conditions. If this happens then the current solution is optimal and the method stops. Otherwise a direction is found alone which the objective function strictly increases. If such a direction is a direction of P then the method stops with the proof that the problem is unbounded.
Otherwise the method moves along d until a new vertex is found with a greater objective value and this iteration is repeated until either an optimal vertex is found or the problem is proved to be unbounded.