# Question: Consider the primal and dual problems in our standard form

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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