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# sec20_Notes - 20 Solving integer programming problems We...

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20 Solving integer programming problems We now have a systematic understanding of how to solve linear programming problems. However, we have so far developed no ideas about how to solve the kinds of integer programming problems that we encountered earlier. The only exception is the special case of transportation problems, where the integrality restriction on the variables turns out to be essentially no restriction at all. Unlike transportation problems (and a few other special problem classes), most integer programming problems are much harder to solve than linear programming problems. In this section we will start to outline a general computational approach. Consider the following example of an integer programming problem with binary variables. maximize 3 x 1 + 2 x 2 + 4 x 3 subject to x 1 + 2 x 2 + x 3 1 x 1 + 2 x 2 + x 3 2 x 1 + x 2 - 2 x 3 0 x 1 , x 2 , x 3 { 0 , 1 } . One foolproof approach is complete enumeration : we simply list all the pos- sible solutions, check which are feasible, and among these, find the best. In this case, we arrive at the following enumeration. x 1 x 2 x 3 Feasible Objective 0 0 0 no 0 0 1 no 0 1 0 yes 2 0 1 1 no 1 0 0 yes 3 1 0 1 no 1 1 0 no 1 1 1 no From the table, we see that the optimal solution of the integer programming problem is [1 , 0 , 0] T . For this small example, complete enumeration was simple and effective. In general, however, this approach is clearly not practical for anything but 102

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the smallest problems, because the number of solutions we must enumerate grows exponentially in the number of binary variables. The next natural idea we might try is rounding . We could use the simplex method to solve the linear programming relaxation : that is, the linear pro- gramming problem we obtain by dropping all the integrality restrictions. We might hope that the optimal solution to the integer programming problem is somehow “close” to the optimal solution of the relaxation. Unfortunately, simple examples show that rounding may fail to say any- thing helpful about the integer programming problem. Consider, for example, the simple integer programming problem maximize x 2 subject to - 2 x 1 + 2 x 2 1 2 x 1 + 2 x 2 7 x 1 , x 2 0 , integer . The optimal solution of the linear programming relaxation is [ 3 2 , 2] T : round- ing the fractional component either up or down leads to an infeasible solution. Rounding gives us no obvious information about optimal solutions of the in- teger programming problem (in this case [1 , 1] T and [2 , 1] T ).
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