[Master Index] [Index for PublicToolbox/regutools]

# 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

This function calls

    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

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