% 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, sedumi is solving the dual problem.
sedumi 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 | 6.500e+00  2.563e+00  2.020e-06 | Solved
10/ 10 | 1.549e+00  1.764e-01  5.261e-07 | Solved
10/ 10 | 4.641e-02  1.530e-04  1.514e-07 | Solved
9/ 10 | 2.950e-03  6.398e-07  1.497e-07 | Solved
0/  0 | 0.000e+00  0.000e+00  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.5685

Successive approximation method to be employed.
sedumi 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 | 4.952e+00  1.547e+00  1.195e-07 | Solved
10/ 10 | 1.014e+00  7.468e-02  3.485e-07 | Solved
10/ 10 | 4.831e-02  1.652e-04  1.021e-07 | Solved
10/ 10 | 3.073e-03  6.930e-07  1.012e-07 | Solved
0/  0 | 0.000e+00  0.000e+00  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.5685