RHSPLOT - Label these four 1 3 vectors y 1 ,y 2 ,y 3 ,y 4 ....

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ORIE 3300/5300 Fall 2011 Prof. Bland Revisiting Right-hand-side Parametrics from the Perspective of LP Duality: An Exercise 1. Suppose that we have identified a dual feasible solution y for a linear program- ming problem maximize cx s.t. Ax b, x 0. If some of the entries in b were to change, would y still be dual feasible? Explain. Re-consider the handout “Right-hand-side Parametrics.” Let P ( b 1 ) denote the max- imization problem (with parameter b 1 ) analyzed there. 2. (a) Suppose y = ( y 1 ,y 2 ,y 3 ) is a dual feasible solution for P ( b 1 ). What does this imply about z * ( b 1 )? Explain and give a geometric interpretation. (b) Now look at the specific case of y 1 = (5 , 0 , 10 3 ) . Plot the dual objective function value of y 1 (as a function of b 1 ) versus b 1 , and indicate what it reveals about z * ( b 1 ). 3. From each of the tableaus # 1,2,3,and 4 in the handout give the dual vec- tor y that is the vector of marginal prices (the π vector) from that tableau.
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Unformatted text preview: Label these four 1 3 vectors y 1 ,y 2 ,y 3 ,y 4 . Give the dual objective func-tion value (as a function of b 1 ) for each of y 1 ,y 2 ,y 3 ,y 4 . Call these functions V 1 ( b 1 ) ,V 2 ( b 1 ) ,V 3 ( b 1 ) ,V 4 ( b 1 ). 4. (a) What do the four functions V 1 ( b 1 ) ,V 2 ( b 1 ) ,V 3 ( b 1 ) ,V 4 ( b 1 ) from part (c) and the Weak Duality Theorem tell you about the function z * ( b 1 )? (b) Draw a gure as in part (2b) showing all four functions V 1 ( b 1 ) ,V 2 ( b 1 ) ,V 3 ( b 1 ) ,V 4 ( b 1 ), rather than only one. What does this plot reveal about z * ( b 1 )? 5. There are innitely many dual feasible solutions y for P ( b 1 ). Describe in words a gure analogous to the gure in part (4b) that shows each of the relevant functions V ( b 1 ), one for each dual feasible solution....
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