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Unformatted text preview: CS 435 : Linear Optimization Fall 2008 Lecture 9: Duality Theorem Lecturer: Sundar Vishwanathan Computer Science & Engineering Indian Institute of Technology, Bombay Here is the duality theorem. 1 Duality Theorem 1 If the primal is feasible and has a nite optimum then the dual is also feasible, has a nite optimum and their optimums must be equal. Proof: We rst show that the dual is feasible. This means we can nd a y which satis es A T y = c ; y . What does A T y = c mean? It means we can write the cost function as a linear combination of the rows of A . Rows of A are also the outward normals to the hyperplanes in Ax b . The inequality y means the coe cients should be nonnegative. We showed that at an optimum point of the primal, we can write the cost vector as a nonnegative linear combination of the normals to the de ning hyperplanes. The coe cients of this nonnegative linear combination yield a feasible point in the dual....
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 Summer '09
 ProfSundar
 Computer Science, Operations Research, Linear Programming, Vector Space, Dual problem, nonnegative linear combination

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