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ps9_ee550_fall08

# ps9_ee550_fall08 - UNIVERSITY OF SOUTHERN CALIFORNIA FALL...

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UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008 1 EE 550: Problem Set 9 (Extra Credit) Due: Monday Dec. 1, 2008 I. MULTI-COMMODITY FLOW Draw a feasible set of ﬂows for the routing problem of sending 13 units of data from node 1 to node 4 over the network in Fig. 1 (i.e., λ 14 = 13 ). All links are bidirectional as in class, with symmetric link capacities as shown. Your solution must satisfy the additional constraint that the sum ﬂow over any link is strictly less than the capacity of the link. 1 4 6 5 3 2 4 12 7 2 9 6 3 5 λ 14 Fig. 1. A multi-node networks. II. OPTIMIZATION NOTE In class we considered the following problem: Minimize: N i =1 φ i ( x i ) Subject to: 0 x i C i for all i ∈ { 1 ,...,N } N i =1 x i = λ Suppose that the φ i ( x ) functions are differentiable. We found that an optimal solution ( x * 1 ,...,x * N ) must satisfy the following derivative matching conditions. 1

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ps9_ee550_fall08 - UNIVERSITY OF SOUTHERN CALIFORNIA FALL...

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