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Unformatted text preview: EE364b Prof. S. Boyd EE364b Homework 5 1. Distributed method for bi-commodity network flow problem. We consider a network (directed graph) with n arcs and p nodes, described by the incidence matrix A R p n , where A ij = 1 , if arc j enters node i 1 , if arc j leaves node i , otherwise . Two commodities flow in the network. Commodity 1 has source vector s R p , and commodity 2 has source vector t R p , which satisfy 1 T s = 1 T t = 0. The flow of commodity 1 on arc i is denoted x i , and the flow of commodity 2 on arc i is denoted y i . Each of the flows must satisfy flow conservation, which can be expressed as Ax + s = 0 (for commodity 1), and Ay + t = 0 (for commodity 2). Arc i has associated flow cost i ( x i , y i ), where i : R 2 R is convex. (We can impose constraints such as nonnegativity of the flows by restricting the domain of i to R 2 + .) One natural form for i is a function only the total traffic on the arc, i.e. , (...
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This note was uploaded on 04/09/2010 for the course EE 360B taught by Professor Stephenboyd during the Fall '09 term at Stanford.
- Fall '09