Theorem 1.2. Let x be a primal feasible solution and let u be a dual feasible solution such that complementary slackness holds between x and u. Then x and u are primal optimal and dual …

Theorem 1. If x and y are feasible solutions to P and D respectively and x, y satisfy complementary slackness conditions, then x and y are optimum. Theorem 2. If x and y are …

The Duality Theorem states that the problems (P) and (D) are intimately re- lated. One way to think about the relationship is to create a table of possibilities.

Theorem 3 (Complementary Slackness) Consider an x0and y0, feasible in the primal and dual respectively. That is, Ax0 b and ATy0= c ; y0 0 . Then cTx0= yTb if and only if ( y0)i> 0 ) Aix0= …

The main implication of Theorem 1 is that if x and y are feasible and satisfy the comple-mentary slackness conditions, then they are optimal. This result leads us to the primal-dual algorithm in …

Theorem (Complementary Slackness) Let x be a feasible solution to the primal and y be a feasible solution to the dual where primal max c x Ax b x 0 dual min b y ATy c y 0: Then x is optimal to …

SOLUTION: To use complementary slackness, we compare x with e, and y with s. In looking at x, we see that e1 = e3 = 0, so those inequality constraints are binding (equality). Then …