lsqi

(PublicToolbox/regutools/lsqi.m in BrainStorm 2.0 (Alpha))

Function Synopsis

[x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0)

Help Text

LSQI Least squares minimizaiton with a quadratic inequality constraint.

 [x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0)
 [x_alpha,lambda] = lsqi(U,sm,X,b,alpha,x_0)  ,  sm = [sigma,mu]

 Least squares minimization with a quadratic inequality constraint:
    min || A x - b ||   subject to   || x - x_0 ||     <= alpha
    min || A x - b ||   subject to   || L (x - x_0) || <= alpha
 where x_0 is an initial guess of the solution, and alpha is a
 positive constant.  Requires either the compact SVD of A saved as
 U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X.
 The regularization parameter lambda is also returned.

 If alpha is a vector, then x_alpha is a matrix such that
    x_alpha = [ x_alpha(1), x_alpha(2), ... ] .

 If x_0 is not specified, x_0 = 0 is used.

Cross-Reference Information

heb_new                            C:\BrainStorm_2001\PublicToolbox\regutools\heb_new.m

Listing of function C:\BrainStorm_2001\PublicToolbox\regutools\lsqi.m

function [x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0)
%LSQI Least squares minimizaiton with a quadratic inequality constraint.
%
% [x_alpha,lambda] = lsqi(U,s,V,b,alpha,x_0)
% [x_alpha,lambda] = lsqi(U,sm,X,b,alpha,x_0)  ,  sm = [sigma,mu]
%
% Least squares minimization with a quadratic inequality constraint:
%    min || A x - b ||   subject to   || x - x_0 ||     <= alpha
%    min || A x - b ||   subject to   || L (x - x_0) || <= alpha
% where x_0 is an initial guess of the solution, and alpha is a
% positive constant.  Requires either the compact SVD of A saved as
% U, s, and V, or part of the GSVD of (A,L) saved as U, sm, and X.
% The regularization parameter lambda is also returned.
%
% If alpha is a vector, then x_alpha is a matrix such that
%    x_alpha = [ x_alpha(1), x_alpha(2), ... ] .
%
% If x_0 is not specified, x_0 = 0 is used.

% Reference: T. F. Chan, J. Olkin & D. W. Cooley, "Solving quadratically
% constrained least squares using block box unconstrained solvers",
% BIT 32 (1992), 481-495.
% Extension to the case x_0 ~= 0 by Per Chr. Hansen, UNI-C, 11/20/91.
% Key point: the initial lambda is almost unaffected by x_0 because
% || x_unreg || >> || x_0 ||.

% Per Christian Hansen, UNI-C, 11/22/91.

% Initialization.
[n,p] = size(V); [p,ps] = size(s);
if (min(alpha)<0)
  error('Illegal inequality constraint alpha')
end
if (nargin==5), x_0 = zeros(n,1); end
la  = length(alpha);
x_k = zeros(n,la);            lambda = zeros(1,la);
snz = length(find(s(:,1)>0)); beta   = U'*b;

if (ps == 1)
  xi = beta(1:snz)./s(1:snz); omega = V'*x_0; s2 = s.^2;
  x_unreg = V(:,1:snz)*xi; norm_x_unreg = norm(x_unreg - x_0);
  for k=1:la
    if (norm_x_unreg <= alpha(k))
      x_alpha(:,k) = x_unreg; lambda(k) = 0;
    else
      lambda_0 = s(snz)*sqrt(norm_x_unreg/alpha(k) - 1);
      lambda(k) = heb_new(lambda_0,alpha(k),s,beta,omega);
      e = s./(s2 + lambda(k)^2); f = s.*e;
      x_alpha(:,k) = V(:,1:p)*(e.*beta + (1-f).*omega);
    end
  end
else
  x_u   = V(:,p+1:n)*beta(p+1:n);  ps1   = p-snz+1;
  xi    = beta(ps1:p)./s(ps1:p,1); gamma = s(:,1)./s(:,2);
  omega = V\x_0; omega = omega(1:p);
  x_unreg = V(:,ps1:p)*xi + x_u;
  norm_Lx_unreg = norm(s(ps1:p,2).*(xi - omega(ps1:p)));
  for k=1:la
    if (norm_Lx_unreg <= alpha(k))
      x_alpha(:,k) = x_unreg; lambda(k) = 0;
    else
      lambda_0 = (s(ps1,1)/s(ps1,2))*sqrt(norm_Lx_unreg/alpha(k) - 1);
      lambda(k) = heb_new(lambda_0,alpha(k),s,beta(1:p),omega);
      e = gamma./(gamma.^2 + lambda(k)^2); f = gamma.*e;
      x_alpha(:,k) = V(:,1:p)*(e.*beta(1:p)./s(:,2) + ...
                               (1-f).*s(:,2).*omega) + x_u;
    end
  end
end