hw5sol - EE364b Prof. S. Boyd EE364b Homework 5 1....

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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. , ( x i , y i ) = f i ( x i + y i ), where f i : R R is convex. In this form, however, is not strictly convex, which will complicate things. To avoid these complications, we will assume that i is strictly convex. The problem of choosing the minimum cost flows that satisfy flow conservation can be expressed as minimize n i =1 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 i . (b) Use your algorithm to solve the particular problem instance with i ( x i , y i ) = ( x i + y i ) 2 + ( x 2 i + y 2 i ) , dom i = R 2 + , with = 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 + ( x 2 + y 2 ) + x + y parenrightBig analytically. You might find this function useful....
View Full Document

This note was uploaded on 04/09/2010 for the course EE 360B taught by Professor Stephenboyd during the Fall '09 term at Stanford.

Page1 / 9

hw5sol - EE364b Prof. S. Boyd EE364b Homework 5 1....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online