Lecture18

# Lecture18 - Introduction to Mathematical Programming IE406...

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Introduction to Mathematical Programming IE406 Lecture 18 Dr. Ted Ralphs

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IE406 Lecture 18 1 Reading for This Lecture Bertsimas 7.4-7.6
IE406 Lecture 18 2 Negative Cost Cycle Algorithm Recall the negative cost cycle algorithm from last lecture. The basic algorithm consisted of iteratively saturating unsaturated negative cost cycles. Questions to be answered How do we start the algorithm (ﬁnd a feasible ﬂow)? How do we ﬁnd unsaturated, negative cost cycles? If there are no unsaturated negative cost cycles, is the current solution optimal? Is the algorithm guaranteed to terminate?

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IE406 Lecture 18 3 The Residual Network We are only interested in arcs on which ﬂow can be pushed (forwards or backwards). Corresponding to a given ﬂow f , we build a residual network . For each arc ( i, j ) in the original network, we introduce two arcs into the residual network: an arc ( i, j ) with capacity u ij - f ij and cost c ij , and an arc ( j, i ) with capacity f ij and cost - c ij . We delete arcs that have zero capacity from the network. Flow on the arc ( j, i ) in the residual network can be interpreted as a reduction in the ﬂow on arc ( i, j ) in the original network.
4 Interpreting the Residual Network Consider a feasible ﬂow f and another feasible ﬂow f + ¯ f . The ﬂow increment ¯ f can be associated with a ﬂow ˜ f in the residual network. ¯ f ij > 0 ˜ f ij = ¯ f ij , ¯ f ij < 0 ˜ f ji = - ¯ f ij , In the residual network, all the ﬂows are positive. ˜ f

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Lecture18 - Introduction to Mathematical Programming IE406...

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