Initial import from https://statweb.stanford.edu/~candes/software/l1magic/v1.11

Additional Clean up of Mac dirs and tex generated files

1 files changed, 176 insertions, 0 deletions

diff --git a/Optimization/tvqc_newton.m b/Optimization/tvqc_newton.m
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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) H11p(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 = H11p(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;  
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%