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Lecture7

# Lecture7 - MS&E111 Introduction to Optimization Prof Amin...

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MS&E111 Lecture 7 Introduction to Optimization April 26, 2006 Prof. Amin Saberi Scribed by: Yusuf Ozuysal 1 Duality Theory and Sensitivity Analysis In the previous lecture, we introduced the dual of a linear program. Given a linear program maximize c T x subject to Ax b x 0 (1) Its dual program will be minimize b T y subject to A T y c y 0 (2) In this lecture, we will see that we can gain a lot of intuition from the dual of a linear program and its optimum solution. In particular, we will see that the optimum solution of the dual program can be used for sensitivity analysis. 1.1 Complementary slackness We have already seen that by strong duality theorem, the optimum solution of the dual LP can be used as a certificate of optimality for the solution of the primal program. But this is not the only use of a dual solution. The values of the variables in the optimum dual solution can tell us about the “importance” of their corresponding constraints in the primal program. The next example will make this more clear: Example: Production Management Let us consider a simplified model of an automobile manufacturer that produces cars and trucks. Manufacturing is organized into four departments: sheet metal stamping, engine assembly, automobile assembly, and truck assembly. The capacity of each department is limited. The following table provides the percentages of each department’s monthly capacity that is consumed by constructing a thousand cars or a thousand trucks, respectively:

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Lecture7 - MS&E111 Introduction to Optimization Prof Amin...

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