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# lect12 - ISE 536Fall03 Linear Programming and Extensions...

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ISE 536–Fall03: Linear Programming and Extensions October 13, 2003 Lecture 12: Geometry of the Dual Lecturer: Fernando Ord´o˜nez 1 Review Duality For any linear programming problem ( P ) min c t x s . t . Ax b x 0 there exists a dual problem ( D ) max y t b s . t . y t A c y 0 that satisﬁes: Weak duality: For any x feasible for ( P ) and y feasible for ( D ) we have c t x b t y . Strong duality: If x * is optimal solution for ( P ), then there exists y * optimal solution for ( D ) and c t x * = b t y * . Note: LP duality is a special case of Lagrangian Duality , where problem ( P ) has associated the following Lagrangian function: L ( x, y, s ) = c t x - y t ( Ax - b ) - s t x , y, s 0 and ( P ) min x max y,s 0 L ( x, y, s ) ( D ) max y,s 0 min x L ( x, y, s ) 1

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2 Complimentary Slackness Theorem (Complimentary Slackness) Let x and y be primal and dual feasible solutions. Then
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lect12 - ISE 536Fall03 Linear Programming and Extensions...

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