Subject: Optimisation and Financial Mathematics

Topic: Non-standard Linear Programming Programmings and Duality

5. Consider the following pair of primal-dual linear programs: maximise cx (primal) subject to Ax 0 minimise by (dual) subject to ATY >c y > 0 for c, x ER", b,y ERM and AERmxn. (a) Prove the Complementary Slackness Theorem: Assume that the primal LP has a feasible solution x and the dual LP has a feasible solution y. Then, x is an optimal solution of the primal LP and y is an optimal solution of the dual LP if and only if (1) either xj = () or 0 Write out d, z and D explicitly. 5. Consider the following pair of primal-dual linear programs: maximise cx (primal) subject to Ax 0 minimise by (dual) subject to ATY >c y > 0 for c, x ER", b,y ERM and AERmxn. (a) Prove the Complementary Slackness Theorem: Assume that the primal LP has a feasible solution x and the dual LP has a feasible solution y. Then, x is an optimal solution of the primal LP and y is an optimal solution of the dual LP if and only if (1) either xj = () or 0 Write out d, z and D explicitly