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16 Apr 1998 (Updated )

Analysis and Solution of Discrete Ill-Posed Problems.

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, IMM, 11/20/91.
% Key point: the initial lambda is almost unaffected by x_0 because
% || x_unreg || >> || x_0 ||.

% Per Christian Hansen, IMM, August 6, 2007.

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

% If alpha > || x_LS - x_0 || then return x_LS, otherwise compute
% lambda via Hebden-Newton iteration using a good initial guess.
% The initial guess lambda_0 is a modified version of the one from
% the Chan-Olkin-Cooley paper.
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)*(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
  if (m>=n)
    x_u   = V(:,p+1:n)*beta(p+1:n);
  else
    x_u   = V(:,p+1:m)*beta(p+1:m);
  end
  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))*(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

%------------------------------------------------------------------

function lambda = heb_new(lambda_0,alpha,s,beta,omega)
%HEB_NEW Newton iteration with Hebden model (utility routine for LSQI).
%
% lambda = heb_new(lambda_0,alpha,s,beta,omega)
%
% Uses Newton iteration with a Hebden (rational) model to find the
% solution lambda to the secular equation
%    || L (x_lambda - x_0) || = alpha ,
% where x_lambda is the solution defined by Tikhonov regularization.
%
% The initial guess is lambda_0.
%
% The norm || L (x_lambda - x_0) || is computed via s, beta and omega.
% Here, s holds either the singular values of A, if L = I, or the
% c,s-pairs of the GSVD of (A,L), if L ~= I.  Moreover, beta = U'*b
% and omega is either V'*x_0 or the first p elements of inv(X)*x_0.

% 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, IMM, 11/20/91.

% Per Christian Hansen, IMM, 12/29/97.

% Set defaults.
thr = sqrt(eps);  % Relative stopping criterion.
it_max = 50;      % Max number of iterations.

% Initialization.
if (lambda_0 < 0)
  error('Initial guess lambda_0 must be nonnegative')
end
[p,ps] = size(s);
if (ps==2), mu = s(:,2); s = s(:,1)./s(:,2); end
s2 = s.^2;

% Iterate, using Hebden-Newton iteration, i.e., solve the nonlinear
% problem || L x ||^(-2) - alpha^(-2) = 0.  This version was found
% experimentally to work slighty better than Newton's method for
% alpha-values near || L x^exact ||.
lambda = lambda_0; step = 1; it = 0;
while (abs(step) > thr*lambda & it < it_max), it = it+1;
  e = s./(s2 + lambda^2); f = s.*e;
  if (ps==1)
    Lx = e.*beta - f.*omega;
  else
    Lx = e.*beta - f.*mu.*omega;
  end
  norm_Lx = norm(Lx);
  Lv = lambda^2*Lx./(s2 + lambda^2);
  step = (lambda/4)*(norm_Lx^2 - alpha^2)/(Lv'*Lx); % Newton step.
  step = (norm_Lx^2/alpha^2)*step; % Hebden step.
  lambda = lambda + step;
  if (lambda < 0), lambda = 2*lambda_0; lambda_0 = 2*lambda_0; end
end

% Terminate with an error if too many iterations.
if (abs(step) > thr*lambda), error('Max. number of iterations reached'), end

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