Lecture15

Lecture15 - Introduction to Mathematical Programming IE406...

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Introduction to Mathematical Programming IE406 Lecture 15 Dr. Ted Ralphs

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IE406 Lecture 15 1 Reading for This Lecture Bertsimas 6.1-6.3
IE406 Lecture 15 2 Large-scale Linear Programming Linear programs occuring in practice can be extremely large. For large LPs, the vast majority of the matrix is superﬂuous. To solve a particular instance, we only need Constraints that are binding at optimality. Variables that are basic at optimality. In fact, we really only need constraints that have positive dual values and variables that have positive values at optimality. If we only had these variables and constraints from the start, we could solve very easily. The problem is that we don’t know which variables and constraints these are.

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IE406 Lecture 15 3 Delayed Column Generation In problems with large numbers of variables, there are two main diﬃculties. The time required just to generate the matrix. The time required to calculate the reduced costs each iteration. We can address both of these problem with delayed column generation . Idea Start with a subset of “promising” columns. Solve the LP with just these columns. Price the remaining columns and add those with negative reduced costs. Iterate.
IE406 Lecture 15 4 Automatic Delayed Column Generation In fact, we only need to ﬁnd the column with the most negative reduced cost . This is an optimization problem! If we can solve this optimization problem, then we can solve the LP

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This note was uploaded on 08/06/2008 for the course IE 406 taught by Professor Ralphs during the Fall '08 term at Lehigh University .

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Lecture15 - Introduction to Mathematical Programming IE406...

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