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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 φ i ( x i , y i ), where φ i : R 2 → R is convex. (We can impose constraints such as nonnegativity of the flows by restricting the domain of φ i to R 2 + .) One natural form for φ i is a function only the total traffic on the arc, i.e. , φ ( x i , y i ) = f i ( x i + y i ), where f i : R → R is convex. In this form, however, φ is not strictly convex, which will complicate things. To avoid these complications, we will assume that φ i is strictly convex. The problem of choosing the minimum cost flows that satisfy flow conservation can be expressed as minimize ∑ n i =1 φ i ( x i , y i ) subject to Ax + s = 0 , Ay + t = 0 , with variables x, y ∈ R n . This is the bi-commodity network flow problem . (a) Propose a distributed solution to the bi-commodity flow problem using dual de- composition. Your solution can refer to the conjugate functions φ ∗ i . (b) Use your algorithm to solve the particular problem instance with φ i ( x i , y i ) = ( x i + y i ) 2 + ǫ ( x 2 i + y 2 i ) , dom φ i = R 2 + , with ǫ = 0 . 1. The other data for this problem can be found in bicommodity_data.m . To check that your method works, compute the optimal value p ⋆ , using cvx . For the subgradient updates use a constant stepsize of 0 . 1. Run the algorithm for 200 iterations and plot the dual lower bound versus iteration. With a logarithmic vertical axis, plot the norms of the residuals for each of the two flow conservation equations, versus iteration number, on the same plot. 1 Hint: We have posted a function [x,y] = quad2_min(eps,alpha,beta) , which com- putes ( x ∗ , y ∗ ) = argmin x ≥ ,y ≥ parenleftBig ( x + y ) 2 + ǫ ( x 2 + y 2 ) + αx + βy parenrightBig analytically. You might find this function useful....
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- Fall '09
- Optimization, Yi, Mathematical optimization, lower bound, Convex Optimization, Ellipsoid method