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Unformatted text preview: 1 Large Scale LP Decomposition and Column Generation To motivate the topic of this section, let us first consider the following optimization problem: Max x 1 + x 2 + 2 y 1 + y 2 subject to x 1 + 2 x 2 + 2 y 1 + y 2 40 x 1 + 3 x 2 30 2 x 1 + x 2 20 x 1 , x 2 P y 1 10 y 2 10 y 1 + y 2 15 y 1 , y 2 Q Observe that except for the linking constraint x 1 + 2 x 2 + 2 y 1 + y 2 40 , the model completely decomposes into two independent subproblems over feasible regions P and Q . We would like to exploit the fact that it is easier to solve problems over P , Q independently. To this aim, consider the graphical representation for P , Q . Note that we denote P 1 ,... ,P 4 the extreme points of P and Q 1 ,... ,Q 5 the extreme points of Q . P 1 = (0 , 0) P 2 = (10 , 0) P 3 = (6 , 8) P 4 = (0 , 10) x 1 x 2 P Q 1 = (0 , 0) Q 2 = (10 , 0) Q 3 = (10 , 5) Q 4 = (5 , 10) Q 5 = (0 , 10) y 1 y 2 Q Let us use the geometric representation for P , Q . In particular, we can express any feasible solution x P ( y Q ) as a convex combination of P ( Q ) extreme points. Then For any ( x 1 ,x 2 ) P , we have parenleftbigg x 1 x 2 parenrightbigg = 1 parenleftbigg parenrightbigg + 2 parenleftbigg 10 parenrightbigg + 3 parenleftbigg 6 8 parenrightbigg + 4 parenleftbigg 10 parenrightbigg (1) for some 1 + 2 + 3 + 4 = 1 and i 0 for all i . For any ( y 1 ,y 2 ) Q , we have parenleftbigg y 1 y 2 parenrightbigg = 1 parenleftbigg parenrightbigg + 2 parenleftbigg 10 parenrightbigg + 3 parenleftbigg 10 5 parenrightbigg + 4 parenleftbigg 5 10 parenrightbigg + 5 parenleftbigg 10 parenrightbigg (2) for some 1 + 2 + 3 + 4 + 5 = 1 and j 0 for all j . 1 Solving for x 1 , x 2 , y 1 , y 2 from (1)(2), we get x 1 = 10 2 + 6 3 x 2 = 8 3 + 10 4 y 1 = 10 2 + 10 3 + 5 4 y 2 = 5 3 + 10 4 + 10 5 Let us now substitute these expressions into the original model to reexpress it in terms of , variables. We get: Max 10 2 + 14 3 + 10 4 + 20 2 + 25 3 + 20 4 + 10 5 s.t. 10 2 + 22 3 + 20 4 + 20 2 + 25 3 + 20 4 + 10 5 40 1 + 2 + 3 + 4 = 1 1 + 2 + 3 + 4 + 5 = 1 all i , j We will call the above problem the Master Problem . Observe that the Master Problem has few rows but many columns . This raises the question whether it is possible to start the solution with a limited number of columns and then later in the solution generate additional columns only if needed. Is this possible? The answer is yes, and the method is similar to the revised simplex method....
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 Fall '08
 TODD

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