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# lec12 - IE521 Advanced Optimization Lecture 12 Dr Zeliha...

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IE521 Advanced Optimization Lecture 12 Dr. Zeliha Akc ¸a January 2012 1 / 23

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Large scale Linear Programming I Linear programs occurring in practice can be extremely large. I They may have a large number of variables or constraint. I For large LPs, the constraint matrix A can be too large to store and process. I To solve a particular instance, we only need I Constraints that are binding at optimality. I Variables that are basic at optimality. I Therefore, only need constraints that have positive dual values (binding constraints) and variables that have positive values at optimality (basic variables). I With only these variables and constraints, we could solve the problem very easily. I The problem is: which variables are basic at optimal and which constraints are binding? 2 / 23
Column Generation Procedure I For problems with large numbers of variables , two main difficulties: The time required to generate the constraint matrix A . The time required to calculate the reduced cost of each variable (to prove the optimality) at each iteration. I These difficulties may be overcome with column generation . I Main steps of the procedure is: Generate an initial subset of columns. Solve the LP with just these columns. Search for the remaining columns and add those with negative reduced costs. Iterate these steps until no column is found with negative reduced cost. 3 / 23

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Generic Column Generation Algorithm I We want to solve an LP with a large number of columns. I Consider the following LP: (LP) min X i C c i x i s.t. X i C A i x i = b x 0 where | C | is a very large number. I Consider the restricted problem obtained by considering only the subset of the columns indexed by set I . (R-LP) min X i I c i x i s.t. X i I A i x i = b x 0 4 / 23
Generic Column Generation Algorithm 1. Solve the restricted LP (R-LP) and calculate the optimal dual solution ( p > = c B B - 1 . ) 2. Generate a new column A j with negative reduced cost for the given dual solution ( p > ): satisfying c j - c B B - 1 A j < 0 . This column can be found by solving the column generation subproblem which is also an optimization problem: w * = min a C c a - c > B B - 1 a , where C is the global set of columns. 3. If the minimum value for the problem w * is < 0 , add the new column to the set I and go to Step 1. If w * > = 0 , stop, we have the optimal LP solution. 5 / 23

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Storing the Set of Generated Columns There are some variants of the algorithm: I Only store the basic variable columns and delete the others from set I . I Store all columns that has been generated so far in I .
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