L15_Duality theorems - Finding the optimal solution of the Dual from the optimal tableau of the prim

L15_Duality theorems - Finding the optimal solution of the Dual from the optimal tableau of the prim

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Duality theorems Finding the dual optimal solution from the primal optimal tableau
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In this lecture we shall present the primal and dual problems in matrix form and prove certain results on the feasible and optimal solutions of the primal and dual problems. The material of this lecture is treated in Chapter 7.5 of your Textbook. Dual problem in Matrix form
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Suppose the primal in (Matrix and ) standard form is 0 b 0 X b, X A = , subject to X c = z Dual problem in Matrix form Maximize
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where [ ] n c c c . . 2 1 = C = n x x x . . 2 1 X , , = mn m m n n a a a a a a a a a . . . . . . . . 2 1 2 22 21 1 12 11 A 1 2 . . m b b b = b
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Letting [ ] m y y y . . 2 1 = Y the dual LPP in matrix form becomes Minimize b Y = w subject to Y A c, Y unrestricted in sign
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If the primal LPP is a minimization problem the dual LPP in matrix form becomes Maximize b Y = w subject to Y A c, Y unrestricted in sign Minimize X c = z 0 b 0 X b, X A = , subject to
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Weak Duality Theorem : For any pair of feasible primal and dual solutions ( X , Y ), the value of the objective function in the minimization problem is an upper bound for the value of objective function in the maximization problem. For the optimal pair (X*, Y*), the values of the objective functions
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L15_Duality theorems - Finding the optimal solution of the Dual from the optimal tableau of the prim

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