simple least squares example using cvx

simpleLS.matlab

%% A imple Least Squares problem
% https://en.wikipedia.org/wiki/Least_squares

% We first consider the most basic convex optimization problem, least-squares 
% (also known as linear regression). In a least-squares problem, we seek 
% x \in R^n that minimizes norm(Ax?b)

% Setup parameters
n = 100;
A = randn(2*n,n);
b = randn(2*n,1);

% The least-squares solution x=(A'A)^{-1}A'b is easily computed using the backslash operator:
x_ls = A \ b;

% Using CVX, the same problem can be solved as follows:  
cvx_begin
   variable x(n)
   minimize( norm( A*x-b ) )
cvx_end



%%  Line-by-line explanation:
%{
cvx_begin
  creates a placeholder for the new CVX problem and prepares Matlab to 
  accept variable declarations, constraints, an objective function, and so forth.

variable x(n) 
  declares x to be an optimization variable of dimension n. CVX requires that all problem variables be declared before they are used in the objective function or constraints.

minimize( norm(A*x-b) ) 
  specifies the objective function to be minimized.

cvx_end 
  signals the end of the CVX specification, and causes the problem to be solved.
%} 



%% Comments
%{
When Matlab reaches the cvx_end command, the least-squares problem is solved, 
and the Matlab variable x is overwritten with the solution of the 
least-squares problem. Now x is an ordinary length-n numerical vector, 
identical to what would be obtained in the traditional approach, at least 
to within the accuracy of the solver. In addition, several additional 
Matlab variables are created; for instance:
    cvx_optval 
        contains the value of the objective function;
    cvx_status 
        contains a string describing the status of the calculation

All of these quantities: x, cvx_optval, and cvx_status, etc. may now be 
freely used in other Matlab statements, just like anything else

The complete cvx user guide can be found here: http://cvxr.com/cvx/cvx_usrguide.pdf
The toolbox comes with a complete set of examples as well. 
%}