A Characterization of the Corner-Point Solutions

and the two values in (1, 1) are both nonnegative. Note that for each functional constraint, there is a nonnegative “slack” between the evaluation on the left-hand side and the constant on the right-hand side. Explicitly, these slacks are given by 1 (= 6 − 5), 4 (= 3 − (−1)), 3 (= 5 − 2), and 1 (= 4 − 3), respectively. Now, if we let s1 = 1, s2 = 4, s3 = 3, and s4 = 1, then the augmented solution (x1, x2, s1, s2, s3, s4) = (1, 1, 1, 4, 3, 1) is, by construction, a feasible solution to the new problem. Thus, by appropriately augmenting a given feasible solution to the original problem, we can always construct a corresponding feasible solution for the new problem. Next, consider any feasible solution to the new problem. Let (x1, x2, s1, s2, s3, s4) = (0, 0, 6, 3, 5, 4), for example. This solution is feasible because