function [X,resnorm,residual,exitflag,output,lambda]=lsqlin(C,d,A,b,Aeq,beq,lb,ub,X0,options,varargin)
%LSQLIN Constrained linear least squares.
%   X=LSQLIN(C,d,A,b) attempts to solve the least-squares problem
%
%           min  0.5*(NORM(C*x-d)).^2       subject to    A*x <= b
%            x
%
%   where C is m-by-n.
%
%   X=LSQLIN(C,d,A,b,Aeq,beq) solves the least-squares
%   (with equality constraints) problem:
%
%           min  0.5*(NORM(C*x-d)).^2    subject to 
%            x                               A*x <= b and Aeq*x = beq
%
%   X=LSQLIN(C,d,A,b,Aeq,beq,LB,UB) defines a set of lower and upper
%   bounds on the design variables, X, so that the solution  
%   is in the range LB <= X <= UB. Use empty matrices for 
%   LB and UB if no bounds exist. Set LB(i) = -Inf if X(i) is unbounded 
%   below; set UB(i) = Inf if X(i) is unbounded above.
%
%   X=LSQLIN(C,d,A,b,Aeq,beq,LB,UB,X0) sets the starting point to X0.
%
%   X=LSQLIN(C,d,A,b,Aeq,beq,LB,UB,X0,OPTIONS) minimizes with the default 
%   optimization parameters replaced by values in the structure OPTIONS, an 
%   argument created with the OPTIMSET function.  See OPTIMSET for details. 
%   Used options are Display, Diagnostics, TolFun, LargeScale, MaxIter, 
%   JacobMult, PrecondBandWidth, TypicalX, TolPCG, and MaxPCGIter. Currently, only
%   'final' and 'off' are valid values for the parameter Display ('iter'
%   is not available).
%
%   X=LSQLIN(C,d,A,b,Aeq,beq,LB,UB,X0,OPTIONS,P1,P2,...) passes the 
%   problem-dependent parameters P1,P2,... directly to the JMFUN function
%   when OPTIMSET('JacobMult',JMFUN) is set. JMFUN is provided by the user. 
%   Pass empty matrices for A, b, Aeq, beq, LB, UB, XO, OPTIONS, to use the 
%   default values.
%
%   [X,RESNORM]=LSQLIN(C,d,A,b) returns the value of the squared 2-norm of the
%   residual: norm(C*X-d)^2.
%
%   [X,RESNORM,RESIDUAL]=LSQLIN(C,d,A,b) returns the residual: C*X-d.
%
%   [X,RESNORM,RESIDUAL,EXITFLAG] = LSQLIN(C,d,A,b) returns an EXITFLAG that 
%   describes the exit condition of LSQLIN. Possible values of EXITFLAG and 
%   the corresponding exit conditions are
%  
%     1  LSQLIN converged to a solution X.
%     3  Change in the residual smaller that the specified tolerance.
%     0  Maximum number of iterations exceeded.
%    -2  Problem is infeasible.
%    -4  Ill-conditioning prevents further optimization.
%    -7  Magnitude of search direction became too small; no further progress can 
%         be made.
%
%   [X,RESNORM,RESIDUAL,EXITFLAG,OUTPUT] = LSQLIN(C,d,A,b) returns a structure
%   OUTPUT with the number of iterations taken in OUTPUT.iterations,
%   the type of algorithm used in OUTPUT.algorithm, the number of conjugate
%   gradient iterations (if used) in OUTPUT.cgiterations, a measure of first 
%   order optimality (if used) in OUPUT.firstorderopt, and the exit message in
%   OUTPUT.message.
%
%   [X,RESNORM,RESIDUAL,EXITFLAG,OUTPUT,LAMBDA]=LSQLIN(C,d,A,b) returns the 
%   set of Lagrangian multipliers LAMBDA, at the solution: LAMBDA.ineqlin 
%   for the linear inequalities C, LAMBDA.eqlin for the linear equalities Ceq, 
%   LAMBDA.lower for LB, and LAMBDA.upper for UB.

%   Copyright 1990-2004 The MathWorks, Inc. 
%   $Revision: 1.29.4.6 $  $Date: 2004/12/06 16:36:14 $

defaultopt = struct('Display','final','Diagnostics','off',...
   'TolFun',100*eps,...
   'LargeScale','on','MaxIter',200,...
   'JacobMult',[],... % empty by default
   'PrecondBandWidth',0,'TypicalX','ones(numberOfVariables,1)',...
   'TolPCG',0.1,'MaxPCGIter','max(1,floor(numberOfVariables/2))');

% If just 'defaults' passed in, return the default options in X
if nargin==1 && nargout <= 1 && isequal(C,'defaults')
   X = defaultopt;
   return
end

if nargin < 2
  error('optim:lsqlin:NotEnoughInputs', ...
        'LSQLIN requires two input arguments.')
end

% Handle missing arguments
if nargin < 10, options = [];
   if nargin < 9, X0 = []; 
      if nargin < 8, ub=[]; 
         if nargin < 7, lb = []; 
            if nargin < 6, beq =[]; 
               if nargin < 5, Aeq = [];
                  if nargin < 4, b = [];
                     if nargin < 3, A = [];
                     end, end, end, end, end, end, end, end
                     
% Check for non-double inputs
% SUPERIORFLOAT errors when superior input is neither single nor double;
% We use try-catch to override SUPERIORFLOAT's error message when input
% data type is integer.
try
    dataType = superiorfloat(C,d,A,b,Aeq,beq,lb,ub,X0);
    if ~isequal('double', dataType)
        error('optim:lsqlin:NonDoubleInput', ...
            'LSQLIN only accepts inputs of data type double.')
    end
catch
    error('optim:lsqlin:NonDoubleInput', ...
        'LSQLIN only accepts inputs of data type double.')
end
                     
% Set up constant strings
medium =  'medium-scale: active-set';
large = 'large-scale: trust-region reflective Newton'; 
unconstrained = 'slash';
output.iterations = []; % initialize so that it will be the first field

% Options setup
largescale = isequal(optimget(options,'LargeScale',defaultopt,'fast'),'on');
diagnostics = isequal(optimget(options,'Diagnostics',defaultopt,'fast'),'on');
mtxmpy = optimget(options,'JacobMult',defaultopt,'fast');
if isequal(mtxmpy,'atamult')
   warning('optim:lsqlin:JacobMultNameClash', ...
           ['Potential function name clash with a Toolbox helper function:\n' ...
            'Use a name besides ''atamult'' for your JacobMult function to\n' ...
            'avoid errors or unexpected results.'])
end
switch optimget(options,'Display',defaultopt,'fast')
case {'off','none'}
   verbosity = 0;
case 'iter'
   verbosity = 2;
case 'final'
   verbosity = 1;
otherwise
   verbosity = 1;
end

% Set the constraints up: defaults and check size
[nineqcstr,numberOfVariables]=size(A);
[neqcstr,numberOfVariableseq]=size(Aeq);
ncstr = nineqcstr + neqcstr;

if isempty(C) || isempty(d)
   error('optim:lsqlin:FirstTwoArgsEmpty', ...
         'The first two arguments to LSQLIN cannot be empty matrices.')
else
   numberOfVariables = max([size(C,2),numberOfVariables]); % In case C is empty
end

[rows,cols]=size(C);
if length(d) ~= rows
   error('optim:lsqlin:InvalidCAndD', ...
         'The number of rows in C must be equal to the length of d.')
end

if length(b) ~= size(A,1)
   error('optim:lsqlin:InvalidAAndB', ...
         'The number of rows in A must be equal to the length of b.')
end

if length(beq) ~= size(Aeq,1)
   error('optim:lsqlin:InvalidAeqAndBeq', ...
         'The number of rows in Aeq must be equal to the length of beq.')
end

if ( ~isempty(A)) && (size(A,2) ~= cols) 
    error('optim:lsqlin:CAndA', ...
          'The number of columns in C must be equal to the number of columns in A.')
end

if ( ~isempty(Aeq)) && (size(Aeq,2) ~= cols)
    error('optim:lsqlin:CAndAeq', ...
          'The number of columns in C must be equal to the number of columns in Aeq.')
end

if isempty(X0), X0=zeros(numberOfVariables,1); end
if isempty(A), A=zeros(0,numberOfVariables); end
if isempty(b), b=zeros(0,1); end
if isempty(Aeq), Aeq=zeros(0,numberOfVariables); end
if isempty(beq), beq=zeros(0,1); end


% Set d, b and X to be column vectors
d = d(:);
b = b(:);
beq = beq(:);
X0 = X0(:);

[X0,lb,ub,msg] = checkbounds(X0,lb,ub,numberOfVariables);
if ~isempty(msg)
   exitflag = -2;
   [resnorm,residual,lambda]=deal([]);
   output.iterations = 0;
   output.algorithm = ''; % not known at this stage
   output.firstorderopt=[];
   output.cgiterations =[];
   output.message = msg;
   X=X0; 
   if verbosity > 0
      disp(msg)
   end
   return
end

% Test if C is all zeros or empty
if norm(C,'inf')==0 || isempty(C)
   C=0; 
end

caller = 'lsqlin';

% Test for constraints
if isempty([Aeq;A]) && isempty([beq;b]) && all(isinf([lb;ub]))
    output.algorithm = unconstrained;
    % If interior-point chosen and no A,Aeq and C isn't short and fat, then call sllsbox   
elseif largescale && (isempty([A;Aeq])) && (rows >= cols)
    output.algorithm = large;
else
    if largescale 
        if (rows < cols)
            warning('optim:lsqlin:MoreColsThanRows', ...
                    ['Large-scale method requires at least as many equations as variables\n' ...
                    '    in C matrix; switching to medium-scale method.'])
        else % ~isempty([A;Aeq]))
            warning('optim:lsqlin:LinConstraints', ...
                    ['Large-scale method can handle bound constraints only;\n' ...
                     '    switching to medium-scale method.'])
        end
    end
    output.algorithm = medium;
end


if diagnostics > 0
   % Do diagnostics on information so far
   gradflag = []; hessflag = []; line_search=[];
   constflag = 0; gradconstflag = 0; non_eq=0;non_ineq=0;
   lin_eq=size(Aeq,1); lin_ineq=size(A,1); XOUT=ones(numberOfVariables,1);
   funfcn{1} = [];ff=[]; GRAD=[];HESS=[];
   confcn{1}=[];c=[];ceq=[];cGRAD=[];ceqGRAD=[];
   msg = diagnose('lsqlin',output,gradflag,hessflag,constflag,gradconstflag,...
      line_search,options,defaultopt,XOUT,non_eq,...
      non_ineq,lin_eq,lin_ineq,lb,ub,funfcn,confcn,ff,GRAD,HESS,c,ceq,cGRAD,ceqGRAD);
end


switch output.algorithm;
case 'slash'
   % Disable the warnings about conditioning for singular and
   % nearly singular matrices
   warningstate1 = warning('off', 'MATLAB:nearlySingularMatrix');
   warningstate2 = warning('off', 'MATLAB:singularMatrix');
   X = C\d;
   % Restore the warning states to their original settings
   warning(warningstate1)
   warning(warningstate2)
   residual = C*X-d;
   resnorm = sum(residual.*residual);
   exitflag = 1;
   lambda.lower = [];
   lambda.upper = [];
   lambda.ineqlin = [];
   lambda.eqlin = [];
   output.iterations = 0;
   output.algorithm = unconstrained;
   output.firstorderopt=[];
   output.cgiterations =[];
   output.message = [];
case 'large-scale: trust-region reflective Newton'; 
   [X,resnorm,residual,firstorderopt,iterations,cgiterations,exitflag,lambda,msg]=...
      sllsbox(C,d,lb,ub,X0,verbosity,options,defaultopt,varargin{:});
   output.iterations = iterations;
   output.algorithm = large;   
   output.firstorderopt = firstorderopt;
   output.cgiterations = cgiterations;
   output.message = msg;
case 'medium-scale: active-set'
   if largescale && ( issparse(A) || issparse(C) || issparse(Aeq) )% asked for sparse
      warning('optim:lsqlin:ConvertToFull', ...
              ['This problem formulation not yet available for sparse matrices.\n' ...
               'Converting to full to solve.'])
   end
   [X,lambdaqp,exitflag,output,dum1,dum2,msg]= ...
      qpsub(full(C),d,[full(Aeq);full(A)],[beq;b],lb,ub,X0,neqcstr,verbosity,caller,ncstr,numberOfVariables,options,defaultopt);
   output.algorithm = medium; % qpsub overwrites output with output.iterations 
   residual = C*X-d;
   resnorm = sum(residual.*residual);
end

if isequal(output.algorithm , medium)
   llb = length(lb); 
   lub = length(ub);
   lambda.lower = zeros(llb,1);
   lambda.upper = zeros(lub,1);
   arglb = ~isinf(lb); lenarglb = nnz(arglb);
   argub = ~isinf(ub); lenargub = nnz(argub);
   lambda.eqlin = lambdaqp(1:neqcstr,1);
   lambda.ineqlin = lambdaqp(neqcstr+1:neqcstr+nineqcstr,1);
   lambda.lower(arglb) = lambdaqp(neqcstr+nineqcstr+1:neqcstr+nineqcstr+lenarglb);
   lambda.upper(argub) = lambdaqp(neqcstr+nineqcstr+lenarglb+1:neqcstr+nineqcstr+lenarglb+lenargub);
   output.firstorderopt=[];
   output.cgiterations =[];
   output.message = msg;

   if exitflag == 1 % qpsub terminated successfully
     normalTerminationMsg = sprintf('Optimization terminated.');  
     if verbosity > 0
       disp(normalTerminationMsg)
     end
     if isempty(msg)
       output.message = normalTerminationMsg;
     else
       % append normal termination msg to current output msg
       output.message = sprintf('%s\n%s',msg,normalTerminationMsg);
     end
   else
     output.message = msg;
   end   
end
