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% l1dantzig_pd.m
%
% Solves
% min_x ||x||_1 subject to ||A'(Ax-b)||_\infty <= epsilon
%
% Recast as linear program
% min_{x,u} sum(u) s.t. x - u <= 0
% -x - u <= 0
% A'(Ax-b) - epsilon <= 0
% -A'(Ax-b) - epsilon <= 0
% and use primal-dual interior point method.
%
% Usage: xp = l1dantzig_pd(x0, A, At, b, epsilon, pdtol, pdmaxiter, cgtol, cgmaxiter)
%
% x0 - Nx1 vector, initial point.
%
% A - Either a handle to a function that takes a N vector and returns a K
% vector , or a KxN 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 a K vector and returns an N vector.
% If A is a KxN matrix, At is ignored.
%
% b - Kx1 vector of observations.
%
% epsilon - scalar or Nx1 vector of correlation constraints
%
% 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 = l1dantzig_pd(x0, A, At, b, epsilon, pdtol, pdmaxiter, cgtol, cgmaxiter)
largescale = isa(A,'function_handle');
if (nargin < 6), pdtol = 1e-3; end
if (nargin < 7), pdmaxiter = 50; end
if (nargin < 8), cgtol = 1e-8; end
if (nargin < 9), cgmaxiter = 200; end
N = length(x0);
alpha = 0.01;
beta = 0.5;
mu = 10;
gradf0 = [zeros(N,1); ones(N,1)];
% starting point --- make sure that it is feasible
if (largescale)
if (max( abs(At(A(x0) - b)) - epsilon ) > 0)
disp('Starting point infeasible; using x0 = At*inv(AAt)*y.');
AAt = @(z) A(At(z));
[w, cgres] = cgsolve(AAt, b, cgtol, cgmaxiter, 0);
if (cgres > 1/2)
disp('A*At is ill-conditioned: cannot find starting point');
xp = x0;
return;
end
x0 = At(w);
end
else
if (max(abs(A'*(A*x0 - b)) - epsilon ) > 0)
disp('Starting point infeasible; using x0 = At*inv(AAt)*y.');
opts.POSDEF = true; opts.SYM = true;
[w, hcond] = linsolve(A*A', b, opts);
if (hcond < 1e-14)
disp('A*At is ill-conditioned: cannot find starting point');
xp = x0;
return;
end
x0 = A'*w;
end
end
x = x0;
u = (0.95)*abs(x0) + (0.10)*max(abs(x0));
% set up for the first iteration
if (largescale)
Atr = At(A(x) - b);
else
Atr = A'*(A*x - b);
end
fu1 = x - u;
fu2 = -x - u;
fe1 = Atr - epsilon;
fe2 = -Atr - epsilon;
lamu1 = -(1./fu1);
lamu2 = -(1./fu2);
lame1 = -(1./fe1);
lame2 = -(1./fe2);
if (largescale)
AtAv = At(A(lame1-lame2));
else
AtAv = A'*(A*(lame1-lame2));
end
% sdg = surrogate duality gap
sdg = -[fu1; fu2; fe1; fe2]'*[lamu1; lamu2; lame1; lame2];
tau = mu*(4*N)/sdg;
% residuals
rdual = gradf0 + [lamu1-lamu2 + AtAv; -lamu1-lamu2];
rcent = -[lamu1.*fu1; lamu2.*fu2; lame1.*fe1; lame2.*fe2] - (1/tau);
resnorm = norm([rdual; rcent]);
% iterations
pditer = 0;
done = (sdg < pdtol) | (pditer >= pdmaxiter);
while (~done)
% solve for step direction
w2 = - 1 - (1/tau)*(1./fu1 + 1./fu2);
sig11 = -lamu1./fu1 - lamu2./fu2;
sig12 = lamu1./fu1 - lamu2./fu2;
siga = -(lame1./fe1 + lame2./fe2);
sigx = sig11 - sig12.^2./sig11;
if (largescale)
w1 = -(1/tau)*( At(A(1./fe2-1./fe1)) + 1./fu2 - 1./fu1 );
w1p = w1 - (sig12./sig11).*w2;
hpfun = @(z) At(A(siga.*At(A(z)))) + sigx.*z;
[dx, cgres, cgiter] = cgsolve(hpfun, 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
AtAdx = At(A(dx));
else
w1 = -(1/tau)*( A'*(A*(1./fe2-1./fe1)) + 1./fu2 - 1./fu1 );
w1p = w1 - (sig12./sig11).*w2;
Hp = A'*(A*sparse(diag(siga))*A')*A + diag(sigx);
opts.POSDEF = true; opts.SYM = true;
[dx, hcond] = linsolve(Hp, 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
AtAdx = A'*(A*dx);
end
du = w2./sig11 - (sig12./sig11).*dx;
dlamu1 = -(lamu1./fu1).*(dx-du) - lamu1 - (1/tau)*1./fu1;
dlamu2 = -(lamu2./fu2).*(-dx-du) - lamu2 - (1/tau)*1./fu2;
dlame1 = -(lame1./fe1).*(AtAdx) - lame1 - (1/tau)*1./fe1;
dlame2 = -(lame2./fe2).*(-AtAdx) - lame2 - (1/tau)*1./fe2;
if (largescale)
AtAdv = At(A(dlame1-dlame2));
else
AtAdv = A'*(A*(dlame1-dlame2));
end
% find minimal step size that keeps ineq functions < 0, dual vars > 0
iu1 = find(dlamu1 < 0); iu2 = find(dlamu2 < 0);
ie1 = find(dlame1 < 0); ie2 = find(dlame2 < 0);
ifu1 = find((dx-du) > 0); ifu2 = find((-dx-du) > 0);
ife1 = find(AtAdx > 0); ife2 = find(-AtAdx > 0);
smax = min(1,min([...
-lamu1(iu1)./dlamu1(iu1); -lamu2(iu2)./dlamu2(iu2); ...
-lame1(ie1)./dlame1(ie1); -lame2(ie2)./dlame2(ie2); ...
-fu1(ifu1)./(dx(ifu1)-du(ifu1)); -fu2(ifu2)./(-dx(ifu2)-du(ifu2)); ...
-fe1(ife1)./AtAdx(ife1); -fe2(ife2)./(-AtAdx(ife2)) ]));
s = 0.99*smax;
% backtracking line search
suffdec = 0;
backiter = 0;
while (~suffdec)
xp = x + s*dx; up = u + s*du;
Atrp = Atr + s*AtAdx; AtAvp = AtAv + s*AtAdv;
fu1p = fu1 + s*(dx-du); fu2p = fu2 + s*(-dx-du);
fe1p = fe1 + s*AtAdx; fe2p = fe2 + s*(-AtAdx);
lamu1p = lamu1 + s*dlamu1; lamu2p = lamu2 + s*dlamu2;
lame1p = lame1 + s*dlame1; lame2p = lame2 + s*dlame2;
rdp = gradf0 + [lamu1p-lamu2p + AtAvp; -lamu1p-lamu2p];
rcp = -[lamu1p.*fu1p; lamu2p.*fu2p; lame1p.*fe1p; lame2p.*fe2p] - (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
% setup for next iteration
x = xp; u = up;
Atr = Atrp; AtAv = AtAvp;
fu1 = fu1p; fu2 = fu2p;
fe1 = fe1p; fe2 = fe2p;
lamu1 = lamu1p; lamu2 = lamu2p;
lame1 = lame1p; lame2 = lame2p;
sdg = -[fu1; fu2; fe1; fe2]'*[lamu1; lamu2; lame1; lame2];
tau = mu*(4*N)/sdg;
rdual = rdp;
rcent = -[lamu1.*fu1; lamu2.*fu2; lame1.*fe1; lame2.*fe2] - (1/tau);
resnorm = norm([rdual; rcent]);
pditer = pditer+1;
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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