L09f05 - CO350 Linear Programming Chapter 4 Introduction to...

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CO350 Linear Programming Chapter 4: Introduction to Duality 20th May 2005
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Chapter 4: Introduction to Duality 1 Recap LP in SEF: ( P ) maximize c T x subject to Ax = b x 0 Dual LP: ( D ) minimize b T y subject to A T y c Complementary Slackness (CS) Condition x * j = 0 or m i =1 a ij y * i = c j (or both) for each j A more useful form: x * j = 0 = m i =1 a ij y * i = c j for each j Theorem 4.7 (CS Theorem) Suppose x * feasible for ( P ) and y * feasible for ( D ) . x * optimal for ( P ) and y * optimal for ( D ) ⇐⇒ CS condition holds for x * , y * . Theorem 4.8 (CS Theorem restated) Suppose x * is feasible for ( P ) . x * optimal for ( P ) ⇐⇒ there exists y * feasible for ( D ) such that CS condition holds for x * , y * .
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Chapter 4: Introduction to Duality 2 Complementary Slackness for Other Forms CS condition for general LP (pg 47) AND x * j = 0 or m i =1 a ij y * i = c j for each j y * i = 0 or m j =1 a ij x * j = b i for each i Interpretation for SEF In SEF, we have Ax = b as constraints. For any feasible x * , we always have m j =1 a ij x * j = b i . Therefore, the above CS condition reduces to x * j = 0 or m i =1 a ij y * i = c j for each j Similarly, x j is a free variable = m i =1 a ij y * i = c j is a constraint for dual LP = x * j = 0 or m i =1 a ij y * i = c j is redundant
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Chapter 4: Introduction to Duality 3 Theorem 4.9 [Important]
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