# Question

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.

(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.

(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.

(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.

(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.

(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.

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