Homework

ELEG 820 - Network Optimziation

Get .m files from here

Run MakeSampleTopology

Run MakeConflictGraph

Run MakeIndeSets

 

With this data, find the optimal flow rates.

Note that MakeSampleTopology makes

  1. Destinations – a list of destination.
  2. LinkPaths – a list of links crossed by the path to each destination. LinkPaths{k} is the list of links crossed by the path to destination(k).

 

Step 1. Compute the flow rate:

Given the vector Mu, compute the data rate for each flow. There is one flow for each destination.

 

Step 2. Compute the link rate over each link:

Given the flow rate to each destination, compute the data rate over each link. We will refer to LinkRate(k) to be the required data rate over link k.

 

Step 3. Find the best independent set for a give mu:

  1. Initialize BestIndeBitRate = 0*Mu;
  2. Initialize BestVal = -inf;
  3. for each element of AllIndeSets, k, compute the sum Mu(i)*BitRate(k,i) where BitRate(k,i) is the data rate over link i during independent set k. (Note that there is no data structure BitRate. You can make it on your own, or do this part some other way.

 

Step 4. Make a new Mu:

For each element of Mu, i, set Mu(i) = Mu(i) + StepSize*(LinkRate(i) – BestIndeBitRate(i)). But if the new Mu(i)<0, then set Mu(i)=0;

 

Step 5. Put it all together: (See SubGradFramework.m)

  1. Initialize Mu = ones(size(Links,1),1);
  2. Set StepSize to some small number (e.g., StepSize  = 0.5).
  3. Initialize AllMu=Mu; % this is used to plot mu
  4. for n = 1 : Some Large Number, run steps 1-4 (in that order).
  5. After computing Mu, set AllMu = [AllMu,Mu];
  6. Then use the following code to print out AllMu

if (rem(n,1000)==0)

for i=1:length(Mu)

                  figure(i)

            plot(AllMu(i,:))

                  getframe;

            end

    end

  1. After the for n=1:some large number is finished, print All Mu

 

What to turn in:

  1. Code
  2. discuss a simple example with a very small step size and explain that the mu and flow rates are correct
  3. Plots for different StepSizes and different topologies. Discuss the impact of step size.
  4. Extra credit: use different bit-rates
  5. Also, using class notes and the paper “Toward tractable computation of the capacity of multihop wireless networks” (available at http://udelmodels.eecis.udel.edu/publications1.php) explain how the capacity optimization problems is reduced optimization of the dual. Also, give the gradient of the dual function and related that to the steps above.