% Section 6.2, Boyd & Vandenberghe "Convex Optimization"
% "Just relax: Convex programming methods for subset selection
%  and sparse approximation" by J. A. Tropp
% Written for CVX by Almir Mutapcic - 02/28/06
%
% We consider a set of linear inequalities A*x <= b which are
% feasible. We apply two heuristics to find a sparse point x that
% satisfies these inequalities.
%
% The (standard) l1-norm heuristic for finding a sparse solution is:
%
%   minimize   ||x||_1
%       s.t.   Ax <= b
%
% The log-based heuristic is an iterative method for finding
% a sparse solution, by finding a local optimal point for the problem:
%
%   minimize   sum(log( delta + |x_i| ))
%       s.t.   Ax <= b
%
% where delta is a small threshold value (determines what is close to zero).
% We cannot solve this problem since it is a minimization of a concave
% function and thus it is not a convex problem. However, we can apply
% a heuristic in which we linearize the objective, solve, and re-iterate.
% This becomes a weighted l1-norm heuristic:
%
%   minimize sum( W_i * |x_i| )
%       s.t. Ax <= b
%
% which in each iteration re-adjusts the weights W_i based on the rule:
%
%   W_i = 1/(delta + |x_i|), where delta is a small threshold value
%
% This algorithm is described in papers:
% "An Affine Scaling Methodology for Best Basis Selection"
%  by B. D. Rao and K. Kreutz-Delgado
% "Portfolio optimization with linear and ï¬?xed transaction costs"
%  by M. S. Lobo, M. Fazel, and S. Boyd

% fix random number generator so we can repeat the experiment
seed = 0;
randn('state',seed);
rand('state',seed);

% the threshold value below which we consider an element to be zero
delta = 1e-8;

% problem dimensions (m inequalities in n-dimensional space)
m = 100;
n = 50;

% construct a feasible set of inequalities
% (this system is feasible for the x0 point)
A  = randn(m,n);
x0 = randn(n,1);
b  = A*x0 + rand(m,1);

% l1-norm heuristic for finding a sparse solution
fprintf(1, 'Finding a sparse feasible point using l1-norm heuristic ...')
cvx_begin
variable x_l1(n)
minimize( norm( x_l1, 1 ) )
subject to
A*x_l1 <= b;
cvx_end

% number of nonzero elements in the solution (its cardinality or diversity)
nnz = length(find( abs(x_l1) > delta ));
fprintf(1,['\nFound a feasible x in R^%d that has %d nonzeros ' ...
'using the l1-norm heuristic.\n'],n,nnz);

% iterative log heuristic
NUM_RUNS = 15;
nnzs = [];
W = ones(n,1); % initial weights

disp([char(10) 'Log-based heuristic:']);
for k = 1:NUM_RUNS
cvx_begin quiet
variable x_log(n)
minimize( sum( W.*abs(x_log) ) )
subject to
A*x_log <= b;
cvx_end

% display new number of nonzeros in the solution vector
nnz = length(find( abs(x_log) > delta ));
nnzs = [nnzs nnz];
fprintf(1,'   found a solution with %d nonzeros...\n', nnz);

% adjust the weights and re-iterate
W = 1./(delta + abs(x_log));
end

% number of nonzero elements in the solution (its cardinality or diversity)
nnz = length(find( abs(x_log) > delta ));
fprintf(1,['\nFound a feasible x in R^%d that has %d nonzeros ' ...
'using the log heuristic.\n'],n,nnz);

% plot number of nonzeros versus iteration
plot(1:NUM_RUNS, nnzs, [1 NUM_RUNS],[nnzs(1) nnzs(1)],'--');
axis([1 NUM_RUNS 0 n])
xlabel('iteration'), ylabel('number of nonzeros (cardinality)');
legend('log heuristic','l1-norm heuristic','Location','SouthEast')
Finding a sparse feasible point using l1-norm heuristic ...
Calling sedumi: 200 variables, 100 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 100, order n = 201, dim = 201, blocks = 1
nnz(A) = 5200 + 0, nnz(ADA) = 2650, nnz(L) = 1375
it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
0 :            8.50E-01 0.000
1 :   3.98E+01 3.79E-01 0.000 0.4456 0.9000 0.9000   6.27  1  1  6.6E+00
2 :  -5.85E+00 1.47E-01 0.000 0.3882 0.9000 0.9000   1.78  1  1  1.9E+00
3 :  -2.47E+01 6.78E-02 0.000 0.4609 0.9000 0.9000   1.49  1  1  7.5E-01
4 :  -3.26E+01 2.55E-02 0.000 0.3769 0.9000 0.9000   1.32  1  1  2.5E-01
5 :  -3.46E+01 1.01E-02 0.000 0.3936 0.9000 0.9000   1.13  1  1  9.5E-02
6 :  -3.55E+01 3.53E-03 0.000 0.3513 0.9000 0.9000   1.05  1  1  3.3E-02
7 :  -3.58E+01 1.30E-03 0.000 0.3695 0.9000 0.9000   1.02  1  1  1.2E-02
8 :  -3.59E+01 2.58E-05 0.000 0.0198 0.9258 0.9000   1.01  1  1  1.9E-03
9 :  -3.59E+01 6.19E-06 0.000 0.2397 0.9000 0.0000   1.00  1  1  8.5E-04
10 :  -3.59E+01 1.39E-06 0.000 0.2251 0.9000 0.8569   1.00  1  1  2.0E-04
11 :  -3.59E+01 1.50E-07 0.000 0.1077 0.9138 0.9000   1.00  1  1  2.9E-05
12 :  -3.59E+01 2.89E-08 0.000 0.1924 0.9180 0.9000   1.00  1  1  6.1E-06
13 :  -3.59E+01 3.29E-09 0.000 0.1139 0.9000 0.9112   1.00  1  1  6.8E-07
14 :  -3.59E+01 4.33E-12 0.000 0.0013 0.9990 0.9990   1.00  1  1
iter seconds digits       c*x               b*y
14      0.1   Inf -3.5940776942e+01 -3.5940776942e+01
|Ax-b| =   2.7e-15, [Ay-c]_+ =   8.9E-15, |x|=  7.1e+00, |y|=  1.0e+01

Detailed timing (sec)
Pre          IPM          Post
1.000E-02    6.000E-02    0.000E+00
Max-norms: ||b||=1, ||c|| = 2.494989e+01,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.58293.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +35.9408

Found a feasible x in R^50 that has 44 nonzeros using the l1-norm heuristic.

Log-based heuristic:
found a solution with 44 nonzeros...
found a solution with 44 nonzeros...
found a solution with 43 nonzeros...
found a solution with 43 nonzeros...
found a solution with 41 nonzeros...
found a solution with 40 nonzeros...
found a solution with 39 nonzeros...
found a solution with 39 nonzeros...
found a solution with 37 nonzeros...
found a solution with 37 nonzeros...
found a solution with 37 nonzeros...
found a solution with 37 nonzeros...
found a solution with 37 nonzeros...
found a solution with 37 nonzeros...
found a solution with 37 nonzeros...

Found a feasible x in R^50 that has 37 nonzeros using the log heuristic. 