4 Let x  [x1 x2 x3 s1 s2 s3] be a primal feasible point for the Dakota problem and y  [ y1 y2 y3 e1 e2 e3] be a dual feasible point.

a Multiply the ith constraint (in standard form) of the primal by yi and sum the resulting constraints.

b Multiply the jth dual constraint (in standard form)

by xj and sum them.

6 . 1 1 The Dual Simplex Method 329 c Compute: part

(a) answer minus part

(b) answer.

d Use the part

(c) answer and the Dual Theorem to show that if x is primal optimal and y is dual optimal, then (38) and (39) hold.

e Use the part

(c) answer to show that if (38) and (39)

both hold, then x is primal optimal and y is dual optimal.

(Hint: Look at Lemma 2.)