Consider the primal and dual problems in our standard form presented in matrix notation at the beginning of Sec. 6.1. Use only this definition of the dual problem for a primal problem in this form to prove each of the following results.

(a) If the functional constraints for the primal problem Ax b are changed to Ax

b, the only resulting change in the dual problem is to delete the nonnegativity constraints, y  0. (Hint:

The constraints Ax b are equivalent to the set of constraints Ax b and Ax  b.)

(b) If the functional constraints for the primal problem Ax b are changed to Ax 

b, the only resulting change in the dual problem is that the nonnegativity constraints y  0 are replaced by nonpositivity constraints y 0, where the current dual variables are interpreted as the negative of the original dual variables. (Hint: The constraints Ax  b are equivalent to Ax b.)

(c) If the nonnegativity constraints for the primal problem x  0 are deleted, the only resulting change in the dual problem is to replace the functional constraints yA  c by yA

c. (Hint:

A variable unconstrained in sign can be replaced by the difference of two nonnegative variables.)