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% l1decode_pd.m
%
% Decoding via linear programming.
% Solve
% min_x ||b-Ax||_1 .
%
% Recast as the linear program
% min_{x,u} sum(u) s.t. -Ax - u + y <= 0
% Ax - u - y <= 0
% and solve using primal-dual interior point method.
%
% Usage: xp = l1decode_pd(x0, A, At, y, pdtol, pdmaxiter, cgtol, cgmaxiter)
%
% x0 - Nx1 vector, initial point.
%
% A - Either a handle to a function that takes a N vector and returns a M
% vector, or a MxN matrix. If A is a function handle, the algorithm
% operates in "largescale" mode, solving the Newton systems via the
% Conjugate Gradients algorithm.
%
% At - Handle to a function that takes an M vector and returns an N vector.
% If A is a matrix, At is ignored.
%
% y - Mx1 observed code (M > N).
%
% pdtol - Tolerance for primal-dual algorithm (algorithm terminates if
% the duality gap is less than pdtol).
% Default = 1e-3.
%
% pdmaxiter - Maximum number of primal-dual iterations.
% Default = 50.
%
% cgtol - Tolerance for Conjugate Gradients; ignored if A is a matrix.
% Default = 1e-8.
%
% cgmaxiter - Maximum number of iterations for Conjugate Gradients; ignored
% if A is a matrix.
% Default = 200.
%
% Written by: Justin Romberg, Caltech
% Email: jrom@acm.caltech.edu
% Created: October 2005
%
function xp = l1decode_pd(x0, A, At, y, pdtol, pdmaxiter, cgtol, cgmaxiter)
largescale = isa(A,'function_handle');
if (nargin < 5), pdtol = 1e-3; end
if (nargin < 6), pdmaxiter = 50; end
if (nargin < 7), cgtol = 1e-8; end
if (nargin < 8), cgmaxiter = 200; end
N = length(x0);
M = length(y);
alpha = 0.01;
beta = 0.5;
mu = 10;
gradf0 = [zeros(N,1); ones(M,1)];
x = x0;
if (largescale), Ax = A(x); else Ax = A*x; end
u = (0.95)*abs(y-Ax) + (0.10)*max(abs(y-Ax));
fu1 = Ax - y - u;
fu2 = -Ax + y - u;
lamu1 = -1./fu1;
lamu2 = -1./fu2;
if (largescale), Atv = At(lamu1-lamu2); else Atv = A'*(lamu1-lamu2); end
sdg = -(fu1'*lamu1 + fu2'*lamu2);
tau = mu*2*M/sdg;
rcent = [-lamu1.*fu1; -lamu2.*fu2] - (1/tau);
rdual = gradf0 + [Atv; -lamu1-lamu2];
resnorm = norm([rdual; rcent]);
pditer = 0;
done = (sdg < pdtol)| (pditer >= pdmaxiter);
while (~done)
pditer = pditer + 1;
w2 = -1 - 1/tau*(1./fu1 + 1./fu2);
sig1 = -lamu1./fu1 - lamu2./fu2;
sig2 = lamu1./fu1 - lamu2./fu2;
sigx = sig1 - sig2.^2./sig1;
if (largescale)
w1 = -1/tau*(At(-1./fu1 + 1./fu2));
w1p = w1 - At((sig2./sig1).*w2);
h11pfun = @(z) At(sigx.*A(z));
[dx, cgres, cgiter] = cgsolve(h11pfun, w1p, cgtol, cgmaxiter, 0);
if (cgres > 1/2)
disp('Cannot solve system. Returning previous iterate. (See Section 4 of notes for more information.)');
xp = x;
return
end
Adx = A(dx);
else
w1 = -1/tau*(A'*(-1./fu1 + 1./fu2));
w1p = w1 - A'*((sig2./sig1).*w2);
H11p = A'*(sparse(diag(sigx))*A);
opts.POSDEF = true; opts.SYM = true;
[dx, hcond] = linsolve(H11p, w1p,opts);
if (hcond < 1e-14)
disp('Matrix ill-conditioned. Returning previous iterate. (See Section 4 of notes for more information.)');
xp = x;
return
end
Adx = A*dx;
end
du = (w2 - sig2.*Adx)./sig1;
dlamu1 = -(lamu1./fu1).*(Adx-du) - lamu1 - (1/tau)*1./fu1;
dlamu2 = (lamu2./fu2).*(Adx + du) -lamu2 - (1/tau)*1./fu2;
if (largescale), Atdv = At(dlamu1-dlamu2); else Atdv = A'*(dlamu1-dlamu2); end
% make sure that the step is feasible: keeps lamu1,lamu2 > 0, fu1,fu2 < 0
indl = find(dlamu1 < 0); indu = find(dlamu2 < 0);
s = min([1; -lamu1(indl)./dlamu1(indl); -lamu2(indu)./dlamu2(indu)]);
indl = find((Adx-du) > 0); indu = find((-Adx-du) > 0);
s = (0.99)*min([s; -fu1(indl)./(Adx(indl)-du(indl)); -fu2(indu)./(-Adx(indu)-du(indu))]);
% backtrack
suffdec = 0;
backiter = 0;
while(~suffdec)
xp = x + s*dx; up = u + s*du;
Axp = Ax + s*Adx; Atvp = Atv + s*Atdv;
lamu1p = lamu1 + s*dlamu1; lamu2p = lamu2 + s*dlamu2;
fu1p = Axp - y - up; fu2p = -Axp + y - up;
rdp = gradf0 + [Atvp; -lamu1p-lamu2p];
rcp = [-lamu1p.*fu1p; -lamu2p.*fu2p] - (1/tau);
suffdec = (norm([rdp; rcp]) <= (1-alpha*s)*resnorm);
s = beta*s;
backiter = backiter + 1;
if (backiter > 32)
disp('Stuck backtracking, returning last iterate. (See Section 4 of notes for more information.)')
xp = x;
return
end
end
% next iteration
x = xp; u = up;
Ax = Axp; Atv = Atvp;
lamu1 = lamu1p; lamu2 = lamu2p;
fu1 = fu1p; fu2 = fu2p;
% surrogate duality gap
sdg = -(fu1'*lamu1 + fu2'*lamu2);
tau = mu*2*M/sdg;
rcent = [-lamu1.*fu1; -lamu2.*fu2] - (1/tau);
rdual = rdp;
resnorm = norm([rdual; rcent]);
done = (sdg < pdtol) | (pditer >= pdmaxiter);
disp(sprintf('Iteration = %d, tau = %8.3e, Primal = %8.3e, PDGap = %8.3e, Dual res = %8.3e',...
pditer, tau, sum(u), sdg, norm(rdual)));
if (largescale)
disp(sprintf(' CG Res = %8.3e, CG Iter = %d', cgres, cgiter));
else
disp(sprintf(' H11p condition number = %8.3e', hcond));
end
end
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