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spgl1.m
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spgl1.m
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function [x,r,g,info] = spgl1( A, b, tau, sigma, x, varargin )
%SPGL1 Solve basis pursuit, basis pursuit denoise, and LASSO
%
% [x, r, g, info] = spgl1(A, b, tau, sigma, x0, options)
% [x, r, g, info] = spgl1(A, b, tau [, sigma, x0] [,options])
%
% ---------------------------------------------------------------------
% Solve the basis pursuit denoise (BPDN) problem
%
% (BPDN) minimize ||x||_1 subj to ||Ax-b||_2 <= sigma,
%
% or the l1-regularized least-squares problem
%
% (LASSO) minimize ||Ax-b||_2 subj to ||x||_1 <= tau.
%
% Setting the optional parameter mu uses updated A and b:
%
% A = [A ] and b = [b]
% [sqrt(mu)I] and [0].
% ---------------------------------------------------------------------
%
% INPUTS
% ======
% A is an m-by-n matrix, explicit or an operator.
% If A is a function, then it must have the signature
%
% y = A(x,mode) if mode == 1 then y = A x (y is m-by-1);
% if mode == 2 then y = A'x (y is n-by-1).
%
% 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 - Ax
% g is the gradient, g = -A'r
% info is a structure with the following information:
% .tau final value of tau (see sigma above)
% .rNorm two-norm of the optimal residual
% .rGap relative duality gap (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 multiplications with A
% .nProdAt number of multiplications with A'
%
% 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.
%
% 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
% =======
% Ewout van den Berg ([email protected])
% Michael P. Friedlander ([email protected])
%
% BUGS
% ====
% Please send bug reports or comments to
% Michael P. Friedlander ([email protected])
% Ewout van den Berg ([email protected])
% 15 Apr 07: First version derived from spg.m.
% 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 Mar 13: Reset f = fOld if curvilinear line-search fails.
% Avoid computing the Barzilai-Borwein scaling parameter
% when both line-search algorithms failed.
% 09 Sep 13: Recompute function information at new x when tau decreases.
% Fixed bug in subspace minimization. Thanks to Yang Lei
% for reporting this bug.
% ----------------------------------------------------------------------
% This file is part of SPGL1 (Spectral Projected-Gradient for L1).
%
% Copyright (C) 2007 Ewout van den Berg and Michael P. Friedlander,
% 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 = '2.0';
DATE = '28 Nov 2019';
t0 = tic(); % Start your watches!
%----------------------------------------------------------------------
% Check arguments and determine solver mode
%----------------------------------------------------------------------
if ~exist('x', 'var'), x = []; end
if ~exist('sigma', 'var'), sigma = []; end
if ~exist('tau', 'var'), tau = []; end
% Allow options to start at sigma or x
if (isstruct(sigma) || ischar(sigma))
varargin = {sigma, x, varargin{:}};
sigma = []; x = [];
elseif (isstruct(x) || ischar(x))
varargin = {x, varargin{:}};
x = [];
end
if ((nargin < 2) || isempty(b) || isempty(A))
error('At least two arguments are required');
elseif (isempty(sigma) && ~isempty(tau))
% Single tau mode
singleTau = true;
else
% Root-finding mode
if (isempty(tau)), tau = 0; end
if (isempty(sigma)), sigma = 0; end
singleTau = false;
end
%----------------------------------------------------------------------
% Determine the problem size
%----------------------------------------------------------------------
m = length(b);
nProdA = 0; % Reset the matrix-vector counters and initialize
nProdAt = 0; % the maximum number of products to one, in case
maxMatvec = 1; % we need to call the Aprod function below.
timeProject = 0;
timeMatProd = 0;
% Determine the problem size and check if problem is complex
explicit = ~(isa(A,'function_handle'));
if isnumeric(A)
n = size(A,2);
realx = isreal(A) && isreal(b);
else
% Infer the size of x based on the size of A'*b
if isempty(x)
x = Aprod(b,2);
n = length(x);
realx = isreal(x) && isreal(b);
x = []; % Use default initialization
else
n = length(x);
realx = isreal(x) && isreal(b);
end
end
%----------------------------------------------------------------------
% Apply default parameters where needed
%----------------------------------------------------------------------
options = spgSetParms(varargin{:});
if (isnan(options.iterations))
options.iterations = 10*m;
end
if (~isnan(options.iscomplex))
if (options.iscomplex == 0)
if (realx == false)
warning('SPGL1:Parameters','Detected complex variables, overriding options.iscomplex parameter');
options.iscomplex = 1;
end
else
realx = false;
end
realx = 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(options.weights))
if (any(~isfinite(options.weights)))
error('Entries in options.weights must be finite');
end
if (any(options.weights <= 0))
error('Entries in options.weights must be strictly positive');
end
else
options.weights = 1;
end
%----------------------------------------------------------------------
% Initialize the local variables
%----------------------------------------------------------------------
% Problem parameters
bNorm = norm(b,2);
mu = options.mu;
weights = options.weights;
sigma2 = (sigma^2) / 2;
% Operational variables
stat = 0;
nPrevVals = options.nPrevVals;
lastFv = -inf(nPrevVals,1); % Last m function values.
stepG = 1; % Step length for projected gradient.
% Compute modes
RFMODE_PRIMAL = 0;
rootfindMode = options.rootfindMode;
hybridMode = options.hybridMode;
l1Mode = isequal(options.primal_norm, @NormL1_primal);
% Bounds
stepMin = options.stepMin;
stepMax = options.stepMax;
maxIts = options.iterations;
maxMatvec = max(3,options.maxMatvec);
maxLineErrors = 10; % Maximum number of line-search failures.
% Tolerance levels
bpTol = options.bpTol;
lsTol = options.lsTol;
optTol = options.optTol;
decTol = options.decTol;
projTol = options.projTol;
relgapMinF = options.relgapMinF; % gap / max(relgapMinF, f)
relgapMinR = options.relgapMinR; % rNorm / max(relgapMinR, rNorm)
rootfindTol = options.rootfindTol; % (dual - sigma) / (primal - sigmal)
if (isnan(bpTol)), bpTol = optTol; end
if (isnan(projTol)), projTol = optTol; end
% Runtime statistics
iter = 0;
nLineTot = 0; % Total number of linesearch steps.
nLineErr = 0;
nNewton = 0;
% Maximum runtime
runtimeCheckEvery = 1; % Check runtime every # iterations
% Output configuration
fid = options.fid;
logLevel = options.verbosity;
printTau = false;
history = options.history;
%----------------------------------------------------------------------
% Exit condition constants
%----------------------------------------------------------------------
EXIT_ROOT_FOUND = 1;
EXIT_BPSOL_FOUND = 2;
EXIT_LEAST_SQUARES = 3;
EXIT_OPTIMAL = 4;
EXIT_ITERATIONS = 5;
EXIT_LINE_ERROR = 6;
EXIT_SUBOPTIMAL_BP = 7;
EXIT_MATVEC_LIMIT = 8;
EXIT_RUNTIME = 9;
EXIT_PROJECTION = 10;
% {Success, Status string }
% {-------- ------------------------------------------}
EXIT_STATUS = cell(9,2);
EXIT_STATUS(1,:) = {true, 'EXIT -- Found a root'};
EXIT_STATUS(2,:) = {true, 'EXIT -- Found a BP solution'};
EXIT_STATUS(3,:) = {true, 'EXIT -- Found a least-squares solution'};
EXIT_STATUS(4,:) = {true, 'EXIT -- Optimal solution found'};
EXIT_STATUS(5,:) = {false, 'ERROR EXIT -- Too many iterations'};
EXIT_STATUS(6,:) = {false, 'ERROR EXIT -- Linesearch error'};
EXIT_STATUS(7,:) = {true, 'EXIT -- Found a suboptimal BP solution'};
EXIT_STATUS(8,:) = {true, 'EXIT -- Maximum matrix-vector operations reached'};
EXIT_STATUS(9,:) = {false, 'ERROR EXIT -- Maximum runtime reached'};
EXIT_STATUS(10,:)= {false, 'ERROR EXIT -- Inaccurate projection'};
%----------------------------------------------------------------------
% Initialize iterate history
%----------------------------------------------------------------------
% Pre-allocate iteration info vectors
if (history)
historyXNorm1 = zeros(min(maxIts,10000),1);
historyRNorm2 = zeros(min(maxIts,10000),1);
historyLambda = zeros(min(maxIts,10000),1);
end
%----------------------------------------------------------------------
% Log header
%----------------------------------------------------------------------
printf('\n');
printf(' %s\n',repmat('=',1,67));
printf(' SPGL1 v%s (%s)\n', REVISION, DATE);
printf(' %s\n',repmat('=',1,67));
printf(' %-20s: %8i %5s', 'No. rows', m, '');
printf(' %-20s: %8.2e\n', 'Initial tau', tau);
printf(' %-20s: %8i %5s', 'No. columns', n,'' );
printf(' %-20s: %8.2e\n', '2-norm of b', bNorm );
printf(' %-20s: %8.2e %5s', 'Optimality tol', optTol, '');
if singleTau
printf(' %-20s: %8.2e\n', 'bound on p-norm of x', tau);
else
printf(' %-20s: %8.2e\n', 'Target ||Ax-b||_2', sigma);
end
printf(' %-20s: %8.2e %5s', 'Basis pursuit tol', bpTol, '');
printf(' %-20s: %8i\n', 'Maximum iterations',maxIts );
printf('\n');
if singleTau
logB = ' %5i %13.7e %13.7e %9.2e';
logH = ' %5s %13s %13s %9s\n';
printf(logH,'Iter','||Ax-b||_2','Relative Gap','||A''r||_d');
else
logB = ' %5i %13.7e %13.7e %9.2e %9.2e';
logH = ' %5s %13s %13s %9s %9s %8s\n';
printf(logH,'Iter','||Ax-b||_2','Relative Gap','Rel Error','||A''r||_d','||x||_p');
end
%----------------------------------------------------------------------
% Setup for the first iteration
%----------------------------------------------------------------------
% When sigma >= ||b|| we have x = 0 as the optimal solution.
% Set tau = 0 to short-circuit the main loop.
if (~isempty(sigma) && (bNorm <= sigma))
printf('W: sigma >= ||b||. Exact solution is x = 0.\n');
tau = 0; x = []; singleTau = true;
stat = EXIT_OPTIMAL;
end
% Default initialization for unspecified x
if isempty(x)
x = zeros(n,1);
r = b;
else
% Project x and compute the residual
x = project(x,tau);
r = b - Aprod(x,1); % r = b - Ax
end
% Compute the objective and gradient
f = (r'*r) / 2;
g = - Aprod(r,2); % g = -A'r = A'(Ax-b)
if (mu > 0)
f = f + (mu/2) * (x'*x);
g = g + mu * x;
end
% Required for nonmonotone strategy
lastFv(1) = f;
fBest = f;
xBest = x;
fOld = f;
% Set current maximum dual objective and the corresponding gNorm value
fDualMax = -Inf;
gNormBest = NaN;
% Root-finding parameters
tauNew = tau;
flagFixTau = false; % Make sure tau is no longer updated
flagUpdateTau = false;
flagRequestUpdateTau = false;
% Compute projected gradient direction and initial steplength.
dx = project(x - g, tau) - x;
dxNorm = norm(dx, inf);
if dxNorm < (1 / stepMax)
gStep = stepMax;
else
gStep = min( stepMax, max(stepMin, 1/dxNorm) );
end
%----------------------------------------------------------------------
% Initialization for the hybrid mode
%----------------------------------------------------------------------
% Make sure we can use the hybrid mode. Generate an error message if the
% hybrid mode does not apply; we could alternatively disable the hybrid
% mode and generate a warning, but this message might be missed.
if (hybridMode)
if ~(realx && l1Mode)
error('SPGL1:Parameters','Hybrid mode only applies to non-complex basis pursuit');
end
end
% Initialize parameter for the hybrid mode
if (hybridMode)
% Determine the initial support
xAbs = abs(x);
xNorm1 = sum(xAbs);
if ((abs(xNorm1 - tau) / max(1,tau)) < 1e-8)
support = false(n,1); % Interior of the L1 ball
else
support = (xAbs >= 1e-9); % Boundary
end
H = []; % Current estimate of the Hessian
sqrt1 = []; % Intermediate square-root values #1
sqrt2 = []; % Intermediate square-root values #2
flagUseHessian = false;
lbfgsHist = options.lbfgsHist;
else
flagUseHessian = false;
end
% Initialize weights when doing l1 minimization with mu > 0.
if ((l1Mode) && (mu > 0))
if (length(weights) == 1)
weightsFull = ones(n,1) * weights;
else
weightsFull = reshape(weights,n,1);
end
end
%----------------------------------------------------------------------
% MAIN LOOP
%----------------------------------------------------------------------
while 1
%------------------------------------------------------------------
% Determine the dual objective
%------------------------------------------------------------------
gNorm = options.dual_norm(-g,weights); % ||-A^T(Ax-b) - mu*x||
rtr = (r'*r);
if ((mu == 0) || (~l1Mode))
% Classic method to determine dual
fDual = r'*b - tau*gNorm - (rtr)/2;
else
% Determine z = |A^Tr| = |-g + mu*x|
z = abs(mu * x - g);
% Solve the subproblem in lambda:
% minimize tau*lambda + (1/2mu)||[z - lambda*w]_+||_2^2
objValue = findLambdaStar(z,weightsFull,tau,mu);
% Compute the dual objective
fDual = r'*b - rtr / 2 - objValue;
end
% Compute the augmented-residual norm
if (mu == 0)
rNorm = norm(r,2);
else
rNorm = sqrt(2*f); % f = 0.5 * r^Tr + 0.5*mu * x^Tx
end
% Use the largest dual objective to determine the relative duality
% gap for the current primal objective
if (fDual > fDualMax)
fDualMax = fDual;
gNormBest = gNorm;
else
fDual = fDualMax;
end
gap = f - fDual;
rGap = abs(gap) / max(relgapMinF,f);
%------------------------------------------------------------------
% Test exit conditions
%------------------------------------------------------------------
% Too many iterations and not converged.
if (iter >= maxIts)
stat = EXIT_ITERATIONS;
end
% Maximum runtime exceeded
if (mod(iter, runtimeCheckEvery) == 0)
% Aim to check every 0.5 seconds, which amounts to a number of
% (0.1 / (runtime / iter)) iterations. We allow increases in the
% check frequency of at most 10. In case the runtime per iterations
% is very high, we check every single iteration. The adaptive
% approach to runtime check frequency is used to reduce the number
% of calls to relatively expensive toc function.
runtime = toc(t0);
runtimeCheckEvery = max(1, min(10*runtimeCheckEvery, floor(0.5*iter / runtime)));
if (runtime > options.maxRuntime)
stat = EXIT_RUNTIME;
end
end
% ====================================================================
% Check optimality conditions
% ====================================================================
% Test if a least-squares solution has been found -- note that this
% will be a sub-optimal solution whenever rNorm is less than sigma;
% it is possible to reduce the norm of x.
if (gNorm <= lsTol * rNorm)
stat = EXIT_LEAST_SQUARES;
end
if (singleTau)
% ==================================================================
% In previous versions the condition was (rNorm < optTol*bNorm). We
% replaced optTol by bpTol for increased control over the stopping
% criteria and to better reflects what the check is for. When the
% condition is met, we exit with EXIT_BPSOL_FOUND. To disable this
% stopping criterion, choose bpTol = 0.
% ==================================================================
if (rNorm < bpTol*bNorm)
stat = EXIT_BPSOL_FOUND;
end
if (rGap <= optTol)
stat = EXIT_OPTIMAL;
end
else
if (rootfindMode == RFMODE_PRIMAL)
% -----------------------------------------
% Primal-based root finding (classic mode)
% -----------------------------------------
aError1 = rNorm - sigma;
aError2 = f - sigma^2 / 2;
rError1 = abs(aError1) / max(1,rNorm);
rError2 = abs(aError2) / max(1,f);
if ((rGap <= max(optTol, rError2)) || (rError1 <= optTol))
if (rNorm <= bpTol * bNorm)
% Residual minimized: basis-pursuit solution
stat = EXIT_BPSOL_FOUND;
elseif (rError1 <= optTol)
% Found an approximate root
stat = EXIT_ROOT_FOUND;
elseif (rNorm <= sigma)
% Found a sub-optimal basis-pursuit solution
stat = EXIT_SUBOPTIMAL_BP;
end
end
% Check if tau needs to be updated
testRelChange1 = (abs(f - fOld) <= decTol * f);
testRelChange2 = (abs(f - fOld) <= 1e-1 * f * (abs(rNorm - sigma)));
flagUpdateTau = (((testRelChange1) && (rNorm > 2 * sigma)) || ...
((testRelChange2) && (rNorm <= 2 * sigma)) || ...
(flagRequestUpdateTau)) && ...
~stat && ~flagUpdateTau;
flagUpdateTau = flagUpdateTau && ~flagFixTau;
flagRequestUpdateTau = false;
if (flagUpdateTau)
tauNew = max(0,tau + (rNorm * aError1) / gNormBest);
end
else
% ------------------------
% Dual-based root finding
% ------------------------
% When the primal objective is sufficiently close to sigma
% it is guaranteed that we can solve the problem for the
% current tau value and we therefore fix tau.
aError1 = rNorm - sigma;
rError1 = abs(aError1) / max(relgapMinR,rNorm);
if (rError1 <= optTol)
flagFixTau = true;
end
% Check the gap (dual - sigma) / (primal - sigmal). The rationale
% for this is that we can take a root-finding step if delta is
% sufficiently far above sigma, relative to the maximum, bounded
% from above by the gap between the primal objective to sigma.
% Note that the function values are the squared values
ratio = (fDual - sigma2) / (f - sigma2);
% The dual objective is used to determine candiate tau values,
% and we maintain the largest one as the solution for the next
% root-finding step.
aErrorDual = (b'*r - tau * gNorm) - rNorm * sigma;
tauNew = max(tauNew, tau + aErrorDual / gNorm);
% Check optimality
if ((flagFixTau) && (rGap <= optTol))
stat = EXIT_ROOT_FOUND;
end
% Root-finding based on the dual - do not update tau if we
% already updated it in the previous iteration.
if ((flagUpdateTau) || (stat) || (flagFixTau))
flagUpdateTau = false;
elseif (rGap <= decTol)
flagUpdateTau = true;
elseif (ratio >= rootfindTol)
flagUpdateTau = true;
end
end
end
%------------------------------------------------------------------
% Update tau if needed
%------------------------------------------------------------------
if (flagUpdateTau)
tauOld = tau;
tau = tauNew;
nNewton = nNewton + 1;
printTau = abs(tauOld - tau) >= 1e-6 * tau; % For log only.
if (tau < tauOld)
% The one-norm ball decreased we need to project it to
% ensure that the next iterate is feasible.
x = project(x,tau);
% Update the residual, gradient, and function value.
r = b - Aprod(x,1); % r = b - Ax
g = - Aprod(r,2); % g = -A'r
f = r'*r / 2;
if (mu > 0)
g = g - mu * x;
f = f + (mu/2) * (x'*x);
end
% Reset the function value history
lastFv = -inf(nPrevVals,1);
lastFv(1) = f;
fBest = f;
xBest = x;
end
% Reset Hessian
if (hybridMode)
H = []; flagUseHessian = false;
end
% Reset status
stat = 0;
% Reset the current maximum dual objective
fDualMax = -Inf;
gNormBest = NaN;
end
%------------------------------------------------------------------
% Print log, update history and act on exit conditions.
%------------------------------------------------------------------
if ((logLevel >= 2) || singleTau || printTau || (iter == 0) || stat)
tauFlag = ' ';
if printTau, tauFlag = sprintf(' %8.2e',tau); end
if singleTau
printf(logB,iter,rNorm,rGap,gNorm);
else
printf(logB,iter,rNorm,rGap,rError1,gNorm);
if printTau
printf(' %s', tauFlag);
end
end
printf('\n');
end
printTau = false;
% Update history info
if (history)
historyXNorm1(iter+1) = options.primal_norm(x,weights);
historyRNorm2(iter+1) = rNorm;
historyLambda(iter+1) = gNorm;
end
if (stat ~= 0), break; end % Act on exit conditions.
%==================================================================
% Iterations begin here
%==================================================================
iter = iter + 1;
xOld = x; fOld = f; gOld = g; rOld = r;
try
% ===================================
% Try a quasi-Newton direction first
% ===================================
if (flagUseHessian)
% --------------------------------------------------
% Step 1. Get search direction
% --------------------------------------------------
if (~any(support))
d = lbfgshprod(H,-g);
else
d = -g;
% Project gradient onto the coefficient space
dTrans = productBMex(signs .* d(support), 1, sqrt1, sqrt2);
% If ||dTrans|| is tiny it is (near) orthogonal to the face
if (norm(dTrans,2) <= 1e-10 * max(1,norm(d,2)))
if (singleTau)
stat = EXIT_OPTIMAL;
else
flagRequestUpdateTau = true;
end
end
if (~lnErr)
% Get quasi-Newton search direction
dQuasi = lbfgshprod(H,dTrans);
% Convert to global domain to get new search direction
d = zeros(n,1);
dSupport = signs .* productBMex(dQuasi, 0, sqrt1, sqrt2);
d(support) = dSupport;
end
end
if (~lnErr)
% --------------------------------------------------
% Step 2. Determine first non-zero entry to hit zero
% --------------------------------------------------
s1 = (((d < 0) & (x > 0)) | ((d > 0) & (x < 0)));
if (any(s1))
gamma = -max(x(s1) ./ d(s1));
else
gamma = +Inf;
end
w = Aprod(d,1);
% --------------------------------------------------
% Step 3. Compute the optimal step length beta
% --------------------------------------------------
enumerator = (w'*r);
denominator = (w'*w);
if (mu > 0)
enumerator = enumerator - mu * (x'*d);
denominator = denominator + mu * (d'*d);
end
beta = enumerator / denominator;
if (beta <= 1e-11)
lnErr = true;
else
beta = min(beta,gamma);
end
end
% --------------------------------------------------
% Check if a suitable step length was found
% --------------------------------------------------
if (~lnErr)
x = xOld + beta * d;
r = r - beta * w; % Avoid evaluating A*x
f = (r'*r) / 2;
if (mu > 0)
f = f + (mu/2) * (x'*x);
end
stepG = beta;
nLineTot = nLineTot + 1;
% Reset function value history after each successful
% hybrid step as a sufficient condition for convergence
lastFv(:) = f;
end
else
% Do not use Hessian
lnErr = true;
end
%---------------------------------------------------------------
% Projected gradient step and linesearch.
%---------------------------------------------------------------
% -----------------------------------------------------------------
% Line search #1: Backtracking curvilinear line search
% -----------------------------------------------------------------
if (lnErr)
[f,x,r,nLine,stepG,lnErr] = ...
spgLineCurvy(x,gStep*g,max(lastFv),@Aprod,b,@project,tau,mu);
nLineTot = nLineTot + nLine;
if (lnErr)
% Projected backtrack failed. Retry with feasible dir'n linesearch.
x = xOld;
f = fOld;
r = rOld;
end
end
% -----------------------------------------------------------------
% Line search #2: Backtracking line search
% -----------------------------------------------------------------
if ((lnErr) && (nLineErr < 5))
nLineErr = nLineErr + 1;
dx = project(x - gStep*g, tau) - x;
Ad = Aprod(dx,1);
% --------------------------------------------------------------
% The objective along the line search is given by f(alpha)
% = 1/2 * ||A(x+alpha*d) - b||_2^2 + mu/2 * ||x + alpha*d||_2^2
% = 1/2 * ||alpha*Ad - r||_2^2 + mu/2 * ||x + alpha*d||_2^2
% = 1/2 * (alpha^2 Ad'*Ad - 2*alpha*real(r'*Ad) + r'*r) +
% mu/2* (x'*x + 2*alpha*real(x'*d) + alpha^2*(d'*d))
% --------------------------------------------------------------
% Find alpha that gives the minimum function values:
% alpha * (Ad'*Ad + mu*d'*d) = real(r'*Ad - mu*x'd)
enumerator = real(r'*Ad);
denominator = (Ad'*Ad);
if (mu > 0)
enumerator = enumerator - mu*real(x'*dx);
denominator = denominator + mu*(dx'*dx);
end
alphaMin = enumerator / denominator;
% Find the maximum alpha value satisfying the Armijo condition:
% f(alpha) <= f(0) + alpha * gamma * real(g'*d)
gamma = 1e-4;
enumerator = enumerator + gamma * dx'*g;
alphaArmijo = 2*enumerator / denominator;
% Choose alpha as the minimum of alphaMin, alphaArmijo, and
% the maximum step lenght of 1 (in which case we reach the
% boundary of the feasible set).
alpha = min([1,alphaMin,alphaArmijo]);
% Update x, the residula, and objective value
if (alpha >= 0)
x = x + alpha * dx;
r = b - Aprod(x,1);
f = (r'*r)/2;
if (mu > 0)
f = f + (mu/2)*(x'*x);
end
lnErr = false;
else
lnErr = true;
end
end
% -----------------------------------------------------------------
% Line search failure: update Barzilai-Borwein scaling factor
% -----------------------------------------------------------------
if (lnErr)
% Revert to previous iterates and damp maximum BB step.
x = xOld;
f = fOld;
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;
nLineErr = 0; % Reset the counter
end
end
% Ensure that the projection is accurate
if (options.primal_norm(x,weights) > tau+projTol)
x = xOld; f = fOld; g = gOld; r = rOld;
stat = EXIT_PROJECTION; break;
end
%---------------------------------------------------------------
% Update gradient and compute new Barzilai-Borwein scaling.
%---------------------------------------------------------------
if (~lnErr)
% Compute the gradient
g = - Aprod(r,2);
if (mu > 0), g = g + mu * x; end
% Compute the differences
s = x - xOld;
y = g - gOld;
sts = s'*s;
sty = s'*y;
if sty <= 0, gStep = stepMax;
else gStep = min( stepMax, max(stepMin, sts/sty) );
end
else
gStep = min(stepMax, gStep);
end
%---------------------------------------------------------------
% Update a Hessian approximation (hybrid mode)
%---------------------------------------------------------------
% Initial checks might fail to detect complex mode when A
% is an opterator, check complexity of x just to be safe.
if (~isreal(x)), hybridMode = false; end
if (hybridMode)
supportOld = support;
absx = abs(x); % Used for support determination later
xNorm1 = sum(absx); % norm(x,1);
% Step 1. Determine support
if ((abs(xNorm1 - tau) / max(1,tau)) > 1e-8)
support = false(n,1); % Interior of the L1 ball
nSupport = n+1;
else
support = (absx > 1e-9);
nSupport = sum(support);
end
% Step 2. Check if the Hessian should be updated
if ((nSupport > 1) && (nSupport <= m) && ...
(iter > 1) && (~lnErr) && ...
(all(support == supportOld)) && ...
(all(sign(x(support)) == sign(xOld(support)))))
% Support remained the same
flagUpdateHessian = true;
else
flagUpdateHessian = false;
end
% Step 3. Check self-projection condition
if (flagUpdateHessian)
if (any(support)) % Boundary of crosspolytope
d = -g;
s1 = ((d < 0) & (x > 0)) | ((d > 0) & (x < 0));
s2 = ((d <= 0) & (x < 0)) | ((d >= 0) & (x > 0));
s3 = ~(s1 | s2); % Zero entries in x
% Compute the absolute sum of the sets
sum1 = sum(abs(d(s1))); % Decreases the one norm
sum2 = sum(abs(d(s2))); % Increases the one norm
sum3 = sum(abs(d(s3))); % Increases the one norm
if (sum1 > sum2+sum3)
% Move into the polytope
flagSelfProj = false;
elseif (sum1 < sum2+sum3)
% Move away from the boundary
flagSelfProj = (max(abs(d(s3))) <= (sum1+sum2) / sum(support));
else
% Move across the face
flagSelfProj = (sum3 == 0);
end
else % Interior of crosspolytope
flagSelfProj = true;
end
if (~flagSelfProj)
flagUpdateHessian = false;
end
end
% Step 4. Reset or update Hessian approximation
if (flagUpdateHessian)