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spgl1General.m
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function [x,r,g,info] = spgl1General(A, b, tau, sigma, x, options, params)
%SPGL1-General: Solves regularized composite programs, including
% a) basis pursuit, basis pursuit denoise and lasso
% b) nonlinear versions of above problems, where forward model is nonlinear
% c) robust penalty on misfit, including huber and students t
%
% [x, r, g, info] = spgl1(A, b, tau, sigma, x, options, params )
%
%
% ---------------------------------------------------------------------
% Solve the regularized composite problem
%
% (GBPDN) minimize ||x|| subj to h(b - f(x)) <= sigma,
%
% or the regularized composite problem
%
% (GLASSO) minimize h(b - f(x)) subj to ||x|| <= tau.
% ---------------------------------------------------------------------
%
% INPUTS
% ======
% A is one of
% (a) an explicit m by n matrix
% (b) an implicit linear operator (SPOT) from R^n to R^m
% (c) a nonlinear function handle from R^n to R^m
%
%
% If A is linear, then it must have the signature
% y = A(x,mode) if mode == 1 then y is m-by-1;
% if mode == 2 then y is n-by-1.
%
% If A is nonlinear, then it must have the signature
% [f] = funForward(x) and
% [gv] = funForward(x, v)
% gv returns the action of the gradient of f on a vector.
%
%
% b is an m-vector.
% tau is a nonnegative scalar; see (LASSO).
% sigma if sigma != inf or != [], then spgl1 will launch into a
% root-finding mode to find the tau above that solves (BPDN).
% In this case, it's STRONGLY recommended that tau = 0.
% x0 is an n-vector estimate of the solution (possibly all
% zeros). If x0 = [], then SPGL1 determines the length n via
% n = length( A'b ) and sets x0 = zeros(n,1).
% options is a structure of options from spgSetParms. Any unset options
% are set to their default value; set options=[] to use all
% default values.
%
% OUTPUTS
% =======
% x is a solution of the problem
% r is the residual, r = b - f(x)
% g is the gradient, g = \nabla h(b - f(x))
% info is a structure with the following information:
% .tau final value of tau (see sigma above)
% .rNorm two-norm of the optimal residual
% .rErr relative error (an optimality measure)
% .gNorm Lagrange multiplier of (LASSO)
% .stat = 1 found a BPDN solution
% = 2 found a BP sol'n; exit based on small gradient
% = 3 found a BP sol'n; exit based on small residual
% = 4 found a LASSO solution
% = 5 error: too many iterations
% = 6 error: linesearch failed
% = 7 error: found suboptimal BP solution
% = 8 error: too many matrix-vector products
% .time total solution time (seconds)
% .nProdA number of function evaluations
% .nProdAt number of gradient evaluations
%
% OPTIONS
% =======
% Use the options structure to control various aspects of the algorithm:
%
% options.fid File ID to direct log output
% .verbosity 0=quiet, 1=some output, 2=more output.
% .iterations Max. number of iterations (default if 10*m).
% .bpTol Tolerance for identifying a basis pursuit solution.
% .optTol Optimality tolerance (default is 1e-4).
% .decTol Larger decTol means more frequent Newton updates.
% .subspaceMin 0=no subspace minimization, 1=subspace minimization.
% .quitPareto 0=normal execution, 1=forces an exit when the pareto curve is reached
% .minPareto Minimum number of spgl1 iterations before checking for quitPareto
% .funPenalty function handle for h(r) alone, with signature
% [f, g] = funPenalty(r)
%
%
% EXAMPLE
% =======
% m = 120; n = 512; k = 20; % m rows, n cols, k nonzeros.
% p = randperm(n); x0 = zeros(n,1); x0(p(1:k)) = sign(randn(k,1));
% A = randn(m,n); [Q,R] = qr(A',0); A = Q';
% b = A*x0 + 0.005 * randn(m,1);
% opts = spgSetParms('optTol',1e-4);
% [x,r,g,info] = spgl1(A, b, 0, 1e-3, [], opts); % Find BP sol'n.
%
% AUTHORS
% ==========================================================
% SPGL1 solver:
% Ewout van den Berg ([email protected])
% Michael P. Friedlander ([email protected])
% Scientific Computing Laboratory (SCL)
% University of British Columbia, Canada.
% ==========================================================
% ==========================================================
% SPGL1 Generalized extension:
% Aleksandr Aravkin ([email protected])
% T.J. Watson Research Center
% ==========================================================
%
% BUGS
% ====
% Please send bug reports or comments to
% Michael P. Friedlander ([email protected])
% Ewout van den Berg ([email protected])
% REVISION HISTORY
% ================
% 15 Apr 07: First version derived from spg.m.
% Michael P. Friedlander ([email protected]).
% Ewout van den Berg ([email protected]).
% 17 Apr 07: Added root-finding code.
% 18 Apr 07: sigma was being compared to 1/2 r'r, rather than
% norm(r), as advertised. Now immediately change sigma to
% (1/2)sigma^2, and changed log output accordingly.
% 24 Apr 07: Added quadratic root-finding code as an option.
% 24 Apr 07: Exit conditions need to guard against small ||r||
% (ie, a BP solution). Added test1,test2,test3 below.
% 15 May 07: Trigger to update tau is now based on relative difference
% in objective between consecutive iterations.
% 15 Jul 07: Added code to allow a limited number of line-search
% errors.
% 23 Feb 08: Fixed bug in one-norm projection using weights. Thanks
% to Xiangrui Meng for reporting this bug.
% 26 May 08: The simple call spgl1(A,b) now solves (BPDN) with sigma=0.
%
% 18 Feb 10: Code branched to SPGL1-SLIM to include custom features:
% -added option to force exit when Pareto curve is reached
% -included ability to handle Kaczmarz operators.
% -hacked out the linesearch fail conditions using a large limit.
% Tim Lin ([email protected]).
% 03 May 10: Added capabilities to work on distributed vectors. Added options for
% parallelized capabilities, max line-search iterations.
% 20 May 10: Made minPareto a user-configurable condition
% 20 Jul 10: Improved performance of spgLine for costly Aprod
% 17 Nov 10: Made default allowance for feasible line-search artificially large to allow for badly scaled problems
% 20 Dec 10: Added option to allow L1-projection failures to issue warning istead of error (ignorePErr)
% 27 Apr 11: Some optimizations via the following:
% -L1 projection on distributed arrays no longer uses sorting
% -disabled restoring to xBest to save memory space
% -disabled subspace minimizaiton permanately to avoid computing active set to save space
% -re-written some expressions to avoid calculating imtermediate results
% 03 May 11: Further memory optimizations, calculated many quantites in-place to avoid temporary allocation of memory, made oneProjector nested
%
% 09 July 12: Redesigned code to solve a more general class of problems,
% including (a) arbitrary differentiable misfits and (b)
% nonlinear forward models.
% Aleksandr Aravkin.
%
% 09 July 12: Removed Kacmarz options (AA).
%
% ----------------------------------------------------------------------
% This file is part of SPGL1 (Spectral Projected-Gradient for L1).
%
% Copyright (C) 2007-2012 Ewout van den Berg, Michael P. Friedlander,
% Aleksandr Aravkin.
% Department of Computer Science, University of British Columbia, Canada.
% All rights reserved. E-mail: <{ewout78,mpf}@cs.ubc.ca>,
%
% SPGL1 is free software; you can redistribute it and/or modify it
% under the terms of the GNU Lesser General Public License as
% published by the Free Software Foundation; either version 2.1 of the
% License, or (at your option) any later version.
%
% SPGL1 is distributed in the hope that it will be useful, but WITHOUT
% ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General
% Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public
% License along with SPGL1; if not, write to the Free Software
% Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301
% USA
% ----------------------------------------------------------------------
REVISION = '$Rev: 84 $';
DATE = '$Date: 2014-03-18 17:41:32 -0700 (Tues, 18 March 2014) $';
REVISION = REVISION(6:end-1);
DATE = DATE(35:55);
% Set to true to display debug flags
PRINT_DEBUG_FLAGS = false;
tic; % Start your watches!
m = length(b);
%----------------------------------------------------------------------
% Check arguments.
%----------------------------------------------------------------------
if ~exist('params', 'var'), params = []; end
if ~exist('options','var'), options = []; end
if ~exist('x','var'), x = []; end
if ~exist('sigma','var'), sigma = []; end
if ~exist('tau','var'), tau = []; end
if nargin < 2 || isempty(b) || isempty(A)
error('At least two arguments are required');
elseif isempty(tau) && isempty(sigma)
tau = 0;
sigma = 0;
singleTau = false;
elseif isempty(sigma) % && ~isempty(tau) <-- implied
singleTau = true;
else
if isempty(tau)
tau = 0;
end
singleTau = false;
end
%----------------------------------------------------------------------
% Grab input options and set defaults where needed.
%----------------------------------------------------------------------
defaultopts = spgSetParms(...
'fid' , 1 , ... % File ID for output
'verbosity' , 2 , ... % Verbosity level
'iterations' , 10*m , ... % Max number of iterations
'nPrevVals' , 3 , ... % Number previous func values for linesearch
'bpTol' , 1e-06 , ... % Tolerance for basis pursuit solution
'lsTol' , 1e-06 , ... % Least-squares optimality tolerance
'optTol' , 1e-04 , ... % Optimality tolerance
'decTol' , 1e-04 , ... % Req'd rel. change in primal obj. for Newton
'stepMin' , 1e-16 , ... % Minimum spectral step
'stepMax' , 1e+05 , ... % Maximum spectral step
'rootMethod' , 2 , ... % Root finding method: 2=quad,1=linear (not used).
'activeSetIt', Inf , ... % Exit with EXIT_ACTIVE_SET if nnz same for # its.
'subspaceMin', 0 , ... % Use subspace minimization
'iscomplex' , NaN , ... % Flag set to indicate complex problem
'maxMatvec' , Inf , ... % Maximum matrix-vector multiplies allowed
'weights' , 1 , ... % Weights W in ||Wx||_1
'quitPareto' , 0 , ... % Exits when pareto curve is reached
'minPareto' , 3 , ... % If quitPareto is on, the minimum number of iterations before checking for quitPareto conditions
'lineSrchIt' , 10, ... % Maximum number of line search iterations for spgLineCurvy, originally 10 ...
'feasSrchIt' , 10 , ... % Maximum number of feasible direction line search iteraitons, originally 10 ...
'ignorePErr' , 0 , ... % Ignores projections error by issuing a warning instead of an error ...
'project' , @NormL1_project , ...
'primal_norm', @NormL1_primal , ...
'dual_norm' , @NormL1_dual , ...
'funPenalty' , @funLS , ... % default penalty - backward compatible with spgl1
'proxy' , 0 , ... % advanced option that computes pareto curve in a user-specified way.
'linear' , 0 , ... % advanced option that allows you to declare input functions to be linear
'restore' , 0 ... % whether to restore best previous answer. for large problems, don't want to do this.
);
options = spgSetParms(defaultopts, options);
fid = options.fid;
logLevel = options.verbosity;
maxIts = options.iterations;
nPrevVals = options.nPrevVals;
bpTol = options.bpTol;
lsTol = options.lsTol;
optTol = options.optTol;
decTol = options.decTol;
stepMin = options.stepMin;
stepMax = options.stepMax;
activeSetIt = options.activeSetIt;
subspaceMin = options.subspaceMin;
maxMatvec = max(3,options.maxMatvec);
weights = options.weights;
maxLineErrors = Inf; % Maximum number of line-search failures (DISABLED)
quitPareto = options.quitPareto;
minPareto = options.minPareto;
lineSrchIt = options.lineSrchIt;
feasSrchIt = options.feasSrchIt;
ignorePErr = options.ignorePErr;
primal_norm = options.primal_norm;
dual_norm = options.dual_norm;
params.proxy = options.proxy;
funPenalty = options.funPenalty;
% definitely don't do subspace minimiation in the non LS case
if(~strcmp(func2str(funPenalty), 'funLS'))
subspaceMin = 0;
end
pivTol = 1e-12; % Threshold for significant Newton step.
%----------------------------------------------------------------------
% Initialize local variables.
%----------------------------------------------------------------------
iter = 0; itnTotLSQR = 0; % Total SPGL1 and LSQR iterations.
nProdA = 0; nProdAt = 0;
lastFv = -inf(nPrevVals,1); % Last m function values.
nLineTot = 0; % Total no. of linesearch steps.
printTau = false;
nNewton = 0;
bNorm = funPenalty(b, params);
stat = false;
timeProject = 0;
timeMatProd = 0;
nnzIter = 0; % No. of its with fixed pattern.
nnzIdx = []; % Active-set indicator.
subspace = false; % Flag if did subspace min in current itn.
stepG = 1; % Step length for projected gradient.
testUpdateTau = 0; % Previous step did not update tau
% Determine initial x, vector length n, and see if problem is complex
explicit = ~(isa(A,'function_handle'));
if isa(A, 'opSpot') || explicit
funForward = @SpotFunForward;
linear = 1;
else
funForward = A;
linear = options.linear;
end
if isempty(x)
if isnumeric(A)
n = size(A,2);
realx = isreal(A) && isreal(b);
else
x = funForward(x, -b);
n = length(x);
realx = isreal(x) && isreal(b);
end
x = zeros(n,1);
else
n = length(x);
realx = isreal(x) && isreal(b);
end
if isnumeric(A), realx = realx && isreal(A); end;
% Override options when options.iscomplex flag is set
if (~isnan(options.iscomplex)), realx = (options.iscomplex == 0); end
% Check if all weights (if any) are strictly positive. In previous
% versions we also checked if the number of weights was equal to
% n. In the case of multiple measurement vectors, this no longer
% needs to apply, so the check was removed.
if ~isempty(weights)
if any(~isfinite(weights))
error('Entries in options.weights must be finite');
end
if any(weights <= 0)
error('Entries in options.weights must be strictly positive');
end
else
weights = 1;
end
% Quick exit if sigma >= ||b||. Set tau = 0 to short-circuit the loop.
if bNorm <= sigma
printf('W: sigma >= ||b||. Exact solution is x = 0.\n');
tau = 0; singleTau = true;
end
% Don't do subspace minimization if x is complex.
if ~realx && subspaceMin
printf('W: Subspace minimization disabled when variables are complex.\n');
subspaceMin = false;
end
% Pre-allocate iteration info vectors
xNorm1 = zeros(min(maxIts,10000),1);
rNorm2 = zeros(min(maxIts,10000),1);
lambda = zeros(min(maxIts,10000),1);
% Exit conditions (constants).
EXIT_ROOT_FOUND = 1;
EXIT_BPSOL_FOUND = 2;
%EXIT_BPSOL2_FOUND = 3;
EXIT_LEAST_SQUARES = 3;
EXIT_OPTIMAL = 4;
EXIT_ITERATIONS = 5;
EXIT_LINE_ERROR = 6;
EXIT_SUBOPTIMAL_BP = 7;
EXIT_MATVEC_LIMIT = 8;
EXIT_ACTIVE_SET = 9; % [sic]
EXIT_AT_PARETO = 10;
%----------------------------------------------------------------------
% Log header.
%----------------------------------------------------------------------
printf('\n');
printf(' %s\n',repmat('=',1,80));
printf(' SPGL1-General v.%s (%s)\n', REVISION, DATE);
printf(' %s\n',repmat('=',1,80));
printf(' %-22s: %8i %4s' ,'No. observations' ,m ,'');
printf(' %-22s: %8i\n' ,'No. unknowns' ,n );
printf(' %-22s: %8.2e %4s' ,'Initial tau' ,tau ,'');
printf(' %-22s: %8s' ,'Penalty ' , func2str(funPenalty));
printf('\n %-22s: %8s' ,'Regularizer' , func2str(primal_norm));
printf(' %-22s: %8.2e\n' ,'Penalty(b)' , bNorm );
printf(' %-22s: %8.2e %4s' ,'Optimality tol' , optTol ,'');
if singleTau
printf(' %-22s: %8.2e\n' ,'Target reg. norm of x' ,tau );
else
printf(' %-22s: %8.2e\n','Target objective' ,sigma );
end
printf(' %-22s: %8.2e %4s' ,'Basis pursuit tol' ,bpTol ,'');
printf(' %-22s: %8i\n' ,'Maximum iterations',maxIts );
printf('\n');
if singleTau
logB = ' %5i %13.7e %13.7e %9.2e %6.1f %6i %6i';
logH = ' %5s %13s %13s %9s %6s %6s %6s\n';
printf(logH,'Iter','Objective','Relative Error','gNorm','stepG','nnzX','nnzG');
else
logB = ' %5i %13.7e %13.7e %9.2e %9.3e %6.1f %6i %6i';
logH = ' %5s %13s %13s %9s %9s %6s %6s %6s %13s\n';
printf(logH,'Iter','Objective','Relative Error','Rel Error',...
'gNorm','stepG','nnzX','nnzG','tau');
end
% Project the starting point and evaluate function and gradient.
if isempty(x)
r = b; % r = b - Ax
[f g g2] = funCompositeR(r, funForward, funPenalty, params);
dx = project(-g, tau);
else
x = project(x,tau);
r = b - funForward(x, [], params); % r = b - f(x)
nProdA = nProdA + 1;
[f g g2] = funCompositeR(r, funForward, funPenalty, params);
dx = project(x - g, tau) - x;
end
% Compute initial steplength.
dxNorm = norm(dx,inf);
if dxNorm < (1 / stepMax)
gStep = stepMax;
else
gStep = min( stepMax, max(stepMin, 1/dxNorm) );
end
% Required for nonmonotone strategy.
lastFv(1) = f;
fBest = f;
xBest = x;
fOld = f;
dispFlag('fin Init')
clear dx;
%----------------------------------------------------------------------
% MAIN LOOP.
%----------------------------------------------------------------------
while 1
%------------------------------------------------------------------
% Test exit conditions.
%------------------------------------------------------------------
% Compute quantities needed for log and exit conditions.
% g2Norm = undist(options.dual_norm(g2,weights,params));
rNorm = f; % rNorm and f are exactly the same.
Err = norm(x - project(x - g, tau));
rErr = Err/max(1, f);
aError1 = rNorm - sigma;
aError2 = rNorm^2 - sigma^2; % Why not?
rError1 = abs(aError1) / max(1,rNorm);
rError2 = abs(aError2) / max(1,f);
% Count number of consecutive iterations with identical support.
nnzOld = nnzIdx;
if(~options.proxy)
[nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzIdx,options, params);
else
% [nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g2,nnzIdx,options, params);
nnzX = 0;
nnzG = 0;
nnzIdx = 0;
nnzDiff = 0;
end
if nnzDiff
nnzIter = 0;
end
dispFlag('fin CompConditions')
if isempty(x)
nnzX = 0;
else
nnzIter = nnzIter + 1;
if nnzIter >= activeSetIt, stat=EXIT_ACTIVE_SET; end
nnzX = sum(abs(x) >= min(.1,10*options.optTol));
end
% Single tau: Check if we're optimal.
% The 2nd condition is there to guard against large tau.
if singleTau
if rErr <= optTol || rNorm < optTol*bNorm
stat = EXIT_OPTIMAL;
end
% Multiple tau: Check if found root and/or if tau needs updating.
else
% Test if a least-squares solution has been found
if rErr <= max(optTol, rError2) || rError1 <= optTol
% The problem is nearly optimal for the current tau.
% Check optimality of the current root.
test1 = rNorm <= bpTol * bNorm;
% test2 = gNorm <= bpTol * rNorm;
test3 = rError1 <= optTol;
test4 = (rNorm <= sigma);% && (abs(rNorm - sigma)<decTol);
% test4 = 0;
if test4, stat=EXIT_SUBOPTIMAL_BP;end % Found suboptimal BP sol.
if test3, stat=EXIT_ROOT_FOUND; end % Found approx root.
% if test2, stat=EXIT_BPSOL2_FOUND; end % Gradient zero -> BP sol.
if test1, stat=EXIT_BPSOL_FOUND; end % Resid minim'zd -> BP sol.
end
testRelChange1 = (abs(f - fOld) <= decTol * f);
testRelChange2 = (abs(f - fOld) <= 1e-2 * (abs(rNorm - sigma)));
testUpdateTau = ((testRelChange1 && rNorm > 2 * sigma) || ...
(testRelChange2 && rNorm <= 2 * sigma)) && ...
~stat && ~testUpdateTau;
if testUpdateTau
if(options.proxy)
gNorm = undist(dual_norm(g2,weights,params)); % originally options.dual_norm(-g,weights), but for true norms the sign should not matter
else
% for now, we assume params only used by proxy formulations
gNorm = undist(dual_norm(g,weights, params)); % originally options.dual_norm(-g,weights), but for true norms the sign should not matter
end
if gNorm <= lsTol % removed '*rNorm'
stat = EXIT_LEAST_SQUARES;
break;
end
if quitPareto && iter >= minPareto, stat=EXIT_AT_PARETO;end % Chose to exit out of SPGL1 when pareto is reached
% Update tau.
tauOld = tau;
tau = max(0,tau + (aError1) / (gNorm)); % deleted rNorm from numerator. In this algorithm, only gNorm with contain derivative information.
nNewton = nNewton + 1;
printTau = abs(tauOld - tau) >= 1e-6 * tau; % For log only.
if tau < tauOld
% The one-norm ball has decreased. Need to make sure that the
% next iterate if feasible, which we do by projecting it.
x = project(x,tau);
r = b - funForward(x, [], params); % r = b - f(x)
nProdA = nProdA + 1;
[f g g2] = funCompositeR(r, funForward, funPenalty, params);
% clear r;
end
end
end
% Too many its and not converged.
if ~stat && iter >= maxIts
stat = EXIT_ITERATIONS;
end
dispFlag('fin CheckConverge')
%------------------------------------------------------------------
% Print log, update history and act on exit conditions.
%------------------------------------------------------------------
if logLevel >= 2 || singleTau || printTau || iter == 0 || stat
tauFlag = ' '; subFlag = '';
if printTau, tauFlag = sprintf(' %13.7e',tau); end
if subspace, subFlag = sprintf(' S %2i',itnLSQR); end
if singleTau
printf(logB,undist(iter),undist(rNorm),undist(rErr),undist(gNorm),log10(undist(stepG)),undist(nnzX),undist(nnzG));
if subspace
printf(' %s',subFlag);
end
else
printf(logB,undist(iter),undist(rNorm),undist(rErr),undist(rError1),undist(gNorm),log10(undist(stepG)),undist(nnzX),undist(nnzG));
if printTau || subspace
printf(' %s',[tauFlag subFlag]);
end
end
printf('\n');
end
printTau = false;
subspace = false;
% Update history info
if isempty(x)
xNorm1(iter+1) = 0;
else
xNorm1(iter+1) = primal_norm(x,weights, params);
end
rNorm2(iter+1) = rNorm;
lambda(iter+1) = gNorm;
if stat, break; end % Act on exit conditions.
%==================================================================
% Iterations begin here.
%==================================================================
iter = iter + 1;
xOld = x; fOld = f; rOld = r;
gOld = g; % gOld update moved down to coincide with gradient update
try
%---------------------------------------------------------------
% Projected gradient step and linesearch.
%---------------------------------------------------------------
dispFlag('begin LineSrch')
[f,x,r,nLine,stepG,lnErr, localProdA] = ...
spgLineCurvy(x,gStep*g,max(lastFv),funForward, funPenalty, b,@project,tau, params);
nProdA = nProdA + localProdA;
dispFlag('fin LineSrch');
nLineTot = nLineTot + nLine;
if lnErr
% Projected backtrack failed. Retry with feasible dir'n linesearch.
dispFlag('begin FeasLineSrch')
clear r
x = xOld;
r = rOld;
f = fOld;
% In-place scaling of gradient and updating of x to save memory
if ~isempty(xOld)
dx = project(xOld - gStep.*g, tau) - xOld;
else
dx = project(-gStep .* g, tau);
end
gtd = dot(g,dx);
if(linear)
[f,step,r,nLine,lnErr, localProdA] = spgLine(f,dx,gtd,rOld,max(lastFv), funForward, funPenalty, params, b,feasSrchIt, linear);
else
[f,step,r,nLine,lnErr, localProdA] = spgLine(f,dx,gtd,x,max(lastFv), funForward, funPenalty, params, b,feasSrchIt, linear);
end
nProdA = nProdA + localProdA;
dispFlag('fin FeasLineSrch')
if isempty(xOld)
x = step*dx;
else
x = xOld + step*dx;
end
clear dx
x = project(x, tau);
nLineTot = nLineTot + nLine;
end
if lnErr
% Failed again. Revert to previous iterates and damp max BB step.
if maxLineErrors <= 0
stat = EXIT_LINE_ERROR;
else
stepMax = stepMax / 10;
printf(['W: Linesearch failed with error %i. '...
'Damping max BB scaling to %6.1e.\n'],lnErr,stepMax);
maxLineErrors = maxLineErrors - 1;
end
end
%---------------------------------------------------------------
% Subspace minimization (only if active-set change is small).
%---------------------------------------------------------------
doSubspaceMin = false;
if subspaceMin
g = - funForward(x, r,2);
[nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzOld,options);
if ~nnzDiff
if nnzX == nnzG, itnMaxLSQR = 20;
else itnMaxLSQR = 5;
end
nnzIdx = abs(x) >= optTol;
doSubspaceMin = true;
end
end
if doSubspaceMin
% LSQR parameters
damp = 1e-5;
aTol = 1e-1;
bTol = 1e-1;
conLim = 1e12;
showLSQR = 0;
ebar = sign(x(nnzIdx));
nebar = length(ebar);
Sprod = @(y,mode)LSQRprod(funForward,nnzIdx,ebar,n,x, y,mode, params);
[dxbar, istop, itnLSQR] = ...
lsqr(m,nebar,Sprod,r,damp,aTol,bTol,conLim,itnMaxLSQR,showLSQR);
itnTotLSQR = itnTotLSQR + itnLSQR;
if istop ~= 4 % LSQR iterations successful. Take the subspace step.
% Push dx back into full space: dx = Z dx.
dx = zeros(n,1);
dx(nnzIdx) = dxbar - (1/nebar)*(ebar'*dxbar)*dxbar;
% Find largest step to a change in sign.
block1 = nnzIdx & x < 0 & dx > +pivTol;
block2 = nnzIdx & x > 0 & dx < -pivTol;
alpha1 = Inf; alpha2 = Inf;
if any(block1), alpha1 = min(-x(block1) ./ dx(block1)); end
if any(block2), alpha2 = min(-x(block2) ./ dx(block2)); end
alpha = min([1 alpha1 alpha2]);
ensure(alpha >= 0);
ensure(ebar'*dx(nnzIdx) <= optTol);
% Update variables.
x = x + alpha*dx;
r = b - funForward(x,[], params);
f = funPenalty(r, params);
subspace = true;
end
end
primNorm_x = undist(primal_norm(x,weights, params));
targetNorm = tau+optTol;
if ignorePErr
if primNorm_x > targetNorm
% warning('Primal norm of projected x is larger than expected, project again to be safe')
% primNorm_x
% targetNorm
end
else
ensure(primNorm_x <= targetNorm);
end
dispFlag('fin UpdateX')
%---------------------------------------------------------------
% Update gradient and compute new Barzilai-Borwein scaling.
%---------------------------------------------------------------
% if isempty(xOld)
% s = x;
% else
% xOld = x - xOld; % in-place calculating of s
% s = xOld;
% clear xOld
% end
% gOld = g;
% g = - Aprod(r,2);
[f g g2] = funCompositeR(r, funForward, funPenalty, params);
if(isempty(xOld))
xOld = x;
else
xOld = x - xOld; % xOld plays the role of s.
end
y = g - gOld;
sts = dot(xOld,xOld);
sty = dot(xOld,y);
clear xOld;
clear gOld;
%[f g g2] = funCompositeR(r, funForward, funPenalty, params);
%[f g g2] = funComposite(x, b, funForward, funPenalty, params); % MAKE SURE X IS CURRENT HERE
% [g ] = funForward(x, r, params);
%nProdAt = nProdAt + 1;
%g = - Aprod(r,2);
%y = g - gOld;
%clear gOld
% sts = norm(s)^2;
% sty = dot(s,y);
if sty <= 0, gStep = stepMax;
else gStep = min( stepMax, max(stepMin, sts/sty) );
end
dispFlag('fin CompScaling')
clear s
clear y
catch % Detect matrix-vector multiply limit error WARNING: the latest round of optimizations may have broke this, do testing to veriify
err = lasterror;
if strcmp(err.identifier,'SPGL1:MaximumMatvec')
stat = EXIT_MATVEC_LIMIT;
iter = iter - 1;
x = xOld; f = fOld; r = rOld;
break;
else
rethrow(err);
end
end
%------------------------------------------------------------------
% Update function history.
%------------------------------------------------------------------
if singleTau || f > sigma % Don't update if superoptimal.
lastFv(mod(iter,nPrevVals)+1) = undist(f);
if fBest > f
fBest = f;
xBest = x;
end
end
end % while 1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Restore best solution (only if solving single problem).
if singleTau && f > fBest
if(options.restore)
rNorm = fBest;
printf('\n Restoring best iterate to objective %13.7e\n',rNorm);
x = xBest;
r = b - funForward(x,[],params);
[f g g2] = funCompositeR(x, r,params);
if(options.proxy)
gNorm = dual_norm(g2,weights);
else
gNorm = dual_norm(g,weights);
end
rNorm = f;
else
printf('NOTE: solution not actually optimal, best objective value is %13.7e',fBest)
end
end
% Final cleanup before exit.
info.tau = tau;
info.rNorm = rNorm;
info.gNorm = gNorm;
info.rErr = rErr;
info.stat = stat;
info.iter = iter;
info.nProdA = nProdA;
info.nProdAt = nProdAt;
info.nNewton = nNewton;
info.timeProject = timeProject;
info.timeMatProd = timeMatProd;
info.itnLSQR = itnTotLSQR;
info.options = options;
info.timeTotal = toc;
info.xNorm1 = xNorm1(1:iter);
info.rNorm2 = rNorm2(1:iter);
info.lambda = lambda(1:iter);
% Print final output.
switch (stat)
case EXIT_OPTIMAL
printf('\n EXIT -- Optimal solution found\n')
case EXIT_ITERATIONS
printf('\n ERROR EXIT -- Too many iterations\n');
case EXIT_ROOT_FOUND
printf('\n EXIT -- Found a root\n');
case {EXIT_BPSOL_FOUND}
printf('\n EXIT -- Found a BP solution\n');
case {EXIT_LEAST_SQUARES}
printf('\n EXIT -- Found a least-squares solution\n');
case EXIT_LINE_ERROR
printf('\n ERROR EXIT -- Linesearch error (%i)\n',lnErr);
case EXIT_SUBOPTIMAL_BP
printf('\n EXIT -- Found a suboptimal BP solution\n');
case EXIT_MATVEC_LIMIT
printf('\n EXIT -- Maximum matrix-vector operations reached\n');
case EXIT_ACTIVE_SET
printf('\n EXIT -- Found a possible active set\n');
case EXIT_AT_PARETO
printf('\n EXIT -- Reached the pareto curve\n');
otherwise
error('Unknown termination condition\n');
end
printf('\n');
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Products with A',nProdA,'','Total time (secs)',info.timeTotal);
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Products with A''',nProdAt,'','Project time (secs)',timeProject);
printf(' %-20s: %6i %6s %-20s: %6.1f\n',...
'Newton iterations',nNewton,'','Mat-vec time (secs)',timeMatProd);
printf(' %-20s: %6i %6s %-20s: %6i\n', ...
'Line search its',nLineTot,'','Subspace iterations',itnTotLSQR);
printf('\n');
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% NESTED FUNCTIONS. These share some vars with workspace above.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% This function is only activated if a spot operator is passed in
function f = SpotFunForward(x, g, params)
if isempty(g)
f = A*x;
else
f = A'*g;
end
end
function [f g1 g2] = funCompositeR(r, funForward, funPenalty, params)
tStart = toc;
nProdAt = nProdAt + 1;
[f v] = funPenalty(r, params);
% f = funPenalty(r, params);
if(~params.proxy)
g1 = funForward(x, -v, params);
% g1 = funForward(x, -r/f, params);
g2 = 0;
else
[g1 g2] = funForward(x, -v, params);
% [g1 g2] = funForward(x, -r/f, params);
end
timeMatProd = timeMatProd + (toc - tStart);
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function printf(varargin)
if logLevel > 0
fprintf(fid,varargin{:});
end
end % function printf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function x = project(x, tau)
dispFlag('begin Project')
tStart = toc;
x = options.project(x,weights,tau, params);
timeProject = timeProject + (toc - tStart);
dispFlag('fin Project')
end % function project
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function dispFlag(flagMsg)
if PRINT_DEBUG_FLAGS
disp(flagMsg)
pause(5)
end
end % function dispFlag
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of nested functions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
end % function spg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PRIVATE FUNCTIONS.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [nnzX,nnzG,nnzIdx,nnzDiff] = activeVars(x,g,nnzIdx,options, params)
% Find the current active set.
% nnzX is the number of nonzero x.