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lecture06_Duality

lecture06_Duality - Yinyu Ye MS&E Stanford Linear...

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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 1 Linear Programming Duality and its Applications Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/˜yyye Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 2 LP example again maximize x 1 +2 x 2 subject to x 1 1 x 2 1 x 1 + x 2 1 . 5 x 1 , x 2 0 . minimize - x 1 - 2 x 2 subject to x 1 + x 3 = 1 x 2 + x 4 = 1 x 1 + x 2 + x 5 = 1 . 5 x 1 , x 2 , x 3 , x 4 , x 5 0 .
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 3 a1 a2 a3 a2 a3 a4 a4 a5 a norm direction cone contained by the norm LP Geometry depicted in two variable space If the direction of c is Objective contour Each corner point has the point is optimal. cone of a corner point, then c Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 4 When a BSF is optimal? Given a BFS in the LP standard form A B x B = b , x B 0 , x N = 0 , and its companion shadow price and reduced cost vectors: A T B y = c B , ( and r = c - A T y ) . If the reduced cost vector r 0 ), then the BFS with basic variable set B , then the BFS optimal . (–What about the maximization ?) At optimality we always have c T x = b T y . What do the shadow prices and reduced costs mean?
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 5 In the LP Example, let B = { 1 , 2 , 3 } so that A B = 1 0 1 0 1 0 1 1 0 , x B = (0 . 5 , 1 , 0 . 5) T and y T = (0 , - 1 , - 1) and r = c - A T y = (0 , 0 , 0 , 1 , 1) T Note that c T x = b T y = - 2 . 5 . Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 6 What do the shadow prices y mean? All inactive constraint have zero shadow price . In general, the shadow price on a given active constraint is the rate of change in the OV as the RHS of the constraint increases , ceteris paribus. If the OS is degenerate , the shadow price may be valid for one-sided changes in the RHS. The constraint RHS ranges give the ranges of the constraint RHS over which no change in the optimal basis will occur. One of the allowable increase and decrease for an inactive constraint is infinite and the other equals to the slack or surplus. In general, when the RHS of an active constraint changes, both the OV and OS will change.
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Yinyu Ye, MS&E, Stanford MS&E211 Lecture Note #06 7 What do the reduced costs r mean? All basic variables have zero reduced cost . In general, the reduced cost coefficient of any non-basic variable is the amount the objective coefficient of that variable would have to change, with all other data held fixed, in order for it to become a basic variable at optimality. If the OS is degenerate , the objective coefficient of a non-basic variable would have to change by at least, and possibly more than, the reduced cost in order to become a basic variable in the OS. The objective coefficient ranges give the ranges of the objective function over which no change in the optimal basis will occur.
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