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% tvqc_newton.m
%
% Newton algorithm for log-barrier subproblems for TV minimization
% with quadratic constraints.
%
% Usage:
% [xp,tp,niter] = tvqc_newton(x0, t0, A, At, b, epsilon, tau,
% newtontol, newtonmaxiter, cgtol, cgmaxiter)
%
% x0,t0 - starting points
%
% 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, constraint relaxation parameter
%
% tau - Log barrier parameter.
%
% newtontol - Terminate when the Newton decrement is <= newtontol.
%
% newtonmaxiter - Maximum number of iterations.
%
% cgtol - Tolerance for Conjugate Gradients; ignored if A is a matrix.
%
% cgmaxiter - Maximum number of iterations for Conjugate Gradients; ignored
% if A is a matrix.
%
% Written by: Justin Romberg, Caltech
% Email: jrom@acm.caltech.edu
% Created: October 2005
%
function [xp, tp, niter] = tvqc_newton(x0, t0, A, At, b, epsilon, tau, newtontol, newtonmaxiter, cgtol, cgmaxiter)
largescale = isa(A,'function_handle');
alpha = 0.01;
beta = 0.5;
N = length(x0);
n = round(sqrt(N));
% create (sparse) differencing matrices for TV
Dv = spdiags([reshape([-ones(n-1,n); zeros(1,n)],N,1) ...
reshape([zeros(1,n); ones(n-1,n)],N,1)], [0 1], N, N);
Dh = spdiags([reshape([-ones(n,n-1) zeros(n,1)],N,1) ...
reshape([zeros(n,1) ones(n,n-1)],N,1)], [0 n], N, N);
if (~largescale), AtA = A'*A; end;
% initial point
x = x0;
t = t0;
if (largescale), r = A(x) - b; else r = A*x - b; end
Dhx = Dh*x; Dvx = Dv*x;
ft = 1/2*(Dhx.^2 + Dvx.^2 - t.^2);
fe = 1/2*(r'*r - epsilon^2);
f = sum(t) - (1/tau)*(sum(log(-ft)) + log(-fe));
niter = 0;
done = 0;
while (~done)
if (largescale), Atr = At(r); else Atr = A'*r; end
ntgx = Dh'*((1./ft).*Dhx) + Dv'*((1./ft).*Dvx) + 1/fe*Atr;
ntgt = -tau - t./ft;
gradf = -(1/tau)*[ntgx; ntgt];
sig22 = 1./ft + (t.^2)./(ft.^2);
sig12 = -t./ft.^2;
sigb = 1./ft.^2 - (sig12.^2)./sig22;
w1p = ntgx - Dh'*(Dhx.*(sig12./sig22).*ntgt) - Dv'*(Dvx.*(sig12./sig22).*ntgt);
if (largescale)
h11pfun = @(z) H11pFun(z, A, At, Dh, Dv, Dhx, Dvx, sigb, ft, fe, Atr);
[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; tp = t;
return
end
Adx = A(dx);
else
H11p = Dh'*sparse(diag(-1./ft + sigb.*Dhx.^2))*Dh + ...
Dv'*sparse(diag(-1./ft + sigb.*Dvx.^2))*Dv + ...
Dh'*sparse(diag(sigb.*Dhx.*Dvx))*Dv + ...
Dv'*sparse(diag(sigb.*Dhx.*Dvx))*Dh - ...
(1/fe)*AtA + (1/fe^2)*Atr*Atr';
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; tp = t;
return
end
Adx = A*dx;
end
Dhdx = Dh*dx; Dvdx = Dv*dx;
dt = (1./sig22).*(ntgt - sig12.*(Dhx.*Dhdx + Dvx.*Dvdx));
% minimum step size that stays in the interior
aqt = Dhdx.^2 + Dvdx.^2 - dt.^2;
bqt = 2*(Dhdx.*Dhx + Dvdx.*Dvx - t.*dt);
cqt = Dhx.^2 + Dvx.^2 - t.^2;
tsols = [(-bqt+sqrt(bqt.^2-4*aqt.*cqt))./(2*aqt); ...
(-bqt-sqrt(bqt.^2-4*aqt.*cqt))./(2*aqt) ];
indt = find([(bqt.^2 > 4*aqt.*cqt); (bqt.^2 > 4*aqt.*cqt)] & (tsols > 0));
aqe = Adx'*Adx; bqe = 2*r'*Adx; cqe = r'*r - epsilon^2;
smax = min(1,min([...
tsols(indt); ...
(-bqe+sqrt(bqe^2-4*aqe*cqe))/(2*aqe)
]));
s = (0.99)*smax;
% backtracking line search
suffdec = 0;
backiter = 0;
while (~suffdec)
xp = x + s*dx; tp = t + s*dt;
rp = r + s*Adx; Dhxp = Dhx + s*Dhdx; Dvxp = Dvx + s*Dvdx;
ftp = 1/2*(Dhxp.^2 + Dvxp.^2 - tp.^2);
fep = 1/2*(rp'*rp - epsilon^2);
fp = sum(tp) - (1/tau)*(sum(log(-ftp)) + log(-fep));
flin = f + alpha*s*(gradf'*[dx; dt]);
suffdec = (fp <= flin);
s = beta*s;
backiter = backiter + 1;
if (backiter > 32)
disp('Stuck on backtracking line search, returning previous iterate. (See Section 4 of notes for more information.)');
xp = x; tp = t;
return
end
end
% set up for next iteration
x = xp; t = tp;
r = rp; Dvx = Dvxp; Dhx = Dhxp;
ft = ftp; fe = fep; f = fp;
lambda2 = -(gradf'*[dx; dt]);
stepsize = s*norm([dx; dt]);
niter = niter + 1;
done = (lambda2/2 < newtontol) | (niter >= newtonmaxiter);
disp(sprintf('Newton iter = %d, Functional = %8.3f, Newton decrement = %8.3f, Stepsize = %8.3e', ...
niter, f, lambda2/2, stepsize));
if (largescale)
disp(sprintf(' CG Res = %8.3e, CG Iter = %d', cgres, cgiter));
else
disp(sprintf(' H11p condition number = %8.3e', hcond));
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% H11p auxiliary function
function y = H11pFun(v, A, At, Dh, Dv, Dhx, Dvx, sigb, ft, fe, atr)
Dhv = Dh*v;
Dvv = Dv*v;
y = Dh'*((-1./ft + sigb.*Dhx.^2).*Dhv + sigb.*Dhx.*Dvx.*Dvv) + ...
Dv'*((-1./ft + sigb.*Dvx.^2).*Dvv + sigb.*Dhx.*Dvx.*Dhv) - ...
1/fe*At(A(v)) + 1/fe^2*(atr'*v)*atr;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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