% Boyd & Vandenberghe "Convex Optimization"
% Argyrios Zymnis - 05/03/08
%
% We consider a network with n flows and L links. Each flow i,
% moves along a fixed predetermined path (i.e. a subset of the links)
% and has an associated rate x_i. Each link j has an associated capacity
% c_j. The total rate of all flows travelling along a link cannot exceed
% the link capacity. We can describe these link capacity limits using the
% flow-link incidence matrix A \in \reals^{L \times n}, where
% A_{ij} = 1, if flow j passes through link i and 0 otherwise.
% The link capacity constraints can be expressed as A*x <= c
% In the network rate problem the variables are the flow rates x. The
% objective is to choose the flow rates to maximize a separate utility
% function U, given by
%           U(x) = U_1(x_1)+U_2(x_2)+...+U_n(x_n)
% The network rate optimization problem is then
%           maximize    U(x)
%           subject to  A*x <= c
% Here we use U_i(x_i) = log x_i for all i

% Input data
rand('state',1)
L = 20;
n = 10;
k = 7; %average links per flow
A = double(rand(L,n) <= k/L);
c = 0.9*rand(L,1)+0.1;

% Solve network rate problem
cvx_begin
    variable x(n);
    maximize(sum(log(x)))
    subject to
        A*x <= c
cvx_end
primal_obj = cvx_optval;

% Solve dual problem to obtain link prices
cvx_begin
    variable lambda(L);
    minimize(c'*lambda-sum(log(A'*lambda))-n)
    subject to
        lambda >= 0
cvx_end
dual_obj = cvx_optval;

 
Successive approximation method to be employed.
   For improved efficiency, SDPT3 is solving the dual problem.
   SDPT3 will be called several times to refine the solution.
   Original size: 50 variables, 20 equality constraints
   10 exponentials add 80 variables, 50 equality constraints
-----------------------------------------------------------------
 Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
 10/ 10 | 6.939e+00  2.563e+00  1.722e-09 | Solved
 10/ 10 | 2.082e+00  2.863e-01  0.000e+00 | Solved
 10/ 10 | 1.575e-01  1.558e-03  0.000e+00 | Solved
 10/ 10 | 2.021e-02  2.577e-05  0.000e+00 | Solved
 10/ 10 | 2.787e-03  4.880e-07  0.000e+00 | Solved
  0/ 10 | 3.880e-04  7.421e-09  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.5685
 
 
Successive approximation method to be employed.
   SDPT3 will be called several times to refine the solution.
   Original size: 50 variables, 20 equality constraints
   10 exponentials add 80 variables, 50 equality constraints
-----------------------------------------------------------------
 Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
 10/ 10 | 5.283e+00  1.547e+00  1.451e-09 | Solved
 10/ 10 | 1.412e+00  1.312e-01  2.742e-09 | Solved
 10/ 10 | 1.306e-01  1.080e-03  2.910e-09 | Solved
 10/ 10 | 1.689e-02  1.815e-05  2.610e-09 | Solved
 10/ 10 | 2.271e-03  3.237e-07  2.635e-09 | Solved
  0/  7 | 3.067e-04  4.651e-09  2.659e-09 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.5685