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Lecture16

# Lecture16 - Advanced Operations Research Techniques IE316...

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Advanced Operations Research Techniques IE316 Lecture 16 Dr. Ted Ralphs

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IE316 Lecture 16 1 Reading for This Lecture Bertsimas 6.1-6.3
IE316 Lecture 16 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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IE316 Lecture 16 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.
IE316 Lecture 16 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 without explicitly listing the columns.

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