comparison pyCSalgos/BP/l1eq_pd.py @ 62:e684f76c1969

Added l1eq_pd(), l1 minimizer with equality constraints from l1-magic

+
+import numpy
+import scipy.linalg
+import math
+
+class l1ecNotImplementedError(Exception):
+  pass
+
+#function xp = l1eq_pd(x0, A, At, b, pdtol, pdmaxiter, cgtol, cgmaxiter)
+def l1eq_pd(x0, A, At, b, pdtol=1e-3, pdmaxiter=50, cgtol=1e-8, cgmaxiter=200, verbose=False):
+
+    # Solve
+    # min_x ||x||_1  s.t.  Ax = b
+    #
+    # Recast as linear program
+    # min_{x,u} sum(u)  s.t.  -u <= x <= u,  Ax=b
+    # and use primal-dual interior point method
+    #
+    # Usage: xp = l1eq_pd(x0, A, At, b, 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.
+    #
+    # 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: [e-mail]
+    # Created: October 2005
+    
+    
+    #---------------------
+    # Original Matab code:    
+        
+    #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);
+    #
+    #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 (norm(A(x0)-b)/norm(b) > cgtol)
+    #    disp('Starting point infeasible; using x0 = At*inv(AAt)*y.');
+    #    AAt = @(z) A(At(z));
+    #    [w, cgres, cgiter] = 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 (norm(A*x0-b)/norm(b) > cgtol)
+    #    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
+    #fu1 = x - u;
+    #fu2 = -x - u;
+    #lamu1 = -1./fu1;
+    #lamu2 = -1./fu2;
+    #if (largescale)
+    #  v = -A(lamu1-lamu2);
+    #  Atv = At(v);
+    #  rpri = A(x) - b;
+    #else
+    #  v = -A*(lamu1-lamu2);
+    #  Atv = A'*v;
+    #  rpri = A*x - b;
+    #end
+    #
+    #sdg = -(fu1'*lamu1 + fu2'*lamu2);
+    #tau = mu*2*N/sdg;
+    #
+    #rcent = [-lamu1.*fu1; -lamu2.*fu2] - (1/tau);
+    #rdual = gradf0 + [lamu1-lamu2; -lamu1-lamu2] + [Atv; zeros(N,1)];
+    #resnorm = norm([rdual; rcent; rpri]);
+    #
+    #pditer = 0;
+    #done = (sdg < pdtol) | (pditer >= pdmaxiter);
+    #while (~done)
+    #  
+    #  pditer = pditer + 1;
+    #  
+    #  w1 = -1/tau*(-1./fu1 + 1./fu2) - Atv;
+    #  w2 = -1 - 1/tau*(1./fu1 + 1./fu2);
+    #  w3 = -rpri;
+    #  
+    #  sig1 = -lamu1./fu1 - lamu2./fu2;
+    #  sig2 = lamu1./fu1 - lamu2./fu2;
+    #  sigx = sig1 - sig2.^2./sig1;
+    #  
+    #  if (largescale)
+    #    w1p = w3 - A(w1./sigx - w2.*sig2./(sigx.*sig1));
+    #    h11pfun = @(z) -A(1./sigx.*At(z));
+    #    [dv, 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
+    #    dx = (w1 - w2.*sig2./sig1 - At(dv))./sigx;
+    #    Adx = A(dx);
+    #    Atdv = At(dv);
+    #  else
+    #    w1p = -(w3 - A*(w1./sigx - w2.*sig2./(sigx.*sig1)));
+    #    H11p = A*(sparse(diag(1./sigx))*A');
+    #    opts.POSDEF = true; opts.SYM = true;
+    #    [dv,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
+    #    dx = (w1 - w2.*sig2./sig1 - A'*dv)./sigx;
+    #    Adx = A*dx;
+    #    Atdv = A'*dv;
+    #  end
+    #  
+    #  du = (w2 - sig2.*dx)./sig1;
+    #  
+    #  dlamu1 = (lamu1./fu1).*(-dx+du) - lamu1 - (1/tau)*1./fu1;
+    #  dlamu2 = (lamu2./fu2).*(dx+du) - lamu2 - 1/tau*1./fu2;
+    #  
+    #  # make sure that the step is feasible: keeps lamu1,lamu2 > 0, fu1,fu2 < 0
+    #  indp = find(dlamu1 < 0);  indn = find(dlamu2 < 0);
+    #  s = min([1; -lamu1(indp)./dlamu1(indp); -lamu2(indn)./dlamu2(indn)]);
+    #  indp = find((dx-du) > 0);  indn = find((-dx-du) > 0);
+    #  s = (0.99)*min([s; -fu1(indp)./(dx(indp)-du(indp)); -fu2(indn)./(-dx(indn)-du(indn))]);
+    #  
+    #  # backtracking line search
+    #  suffdec = 0;
+    #  backiter = 0;
+    #  while (~suffdec)
+    #    xp = x + s*dx;  up = u + s*du; 
+    #    vp = v + s*dv;  Atvp = Atv + s*Atdv; 
+    #    lamu1p = lamu1 + s*dlamu1;  lamu2p = lamu2 + s*dlamu2;
+    #    fu1p = xp - up;  fu2p = -xp - up;  
+    #    rdp = gradf0 + [lamu1p-lamu2p; -lamu1p-lamu2p] + [Atvp; zeros(N,1)];
+    #    rcp = [-lamu1p.*fu1p; -lamu2p.*fu2p] - (1/tau);
+    #    rpp = rpri + s*Adx;
+    #    suffdec = (norm([rdp; rcp; rpp]) <= (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;
+    #  v = vp;  Atv = Atvp; 
+    #  lamu1 = lamu1p;  lamu2 = lamu2p;
+    #  fu1 = fu1p;  fu2 = fu2p;
+    #  
+    #  # surrogate duality gap
+    #  sdg = -(fu1'*lamu1 + fu2'*lamu2);
+    #  tau = mu*2*N/sdg;
+    #  rpri = rpp;
+    #  rcent = [-lamu1.*fu1; -lamu2.*fu2] - (1/tau);
+    #  rdual = gradf0 + [lamu1-lamu2; -lamu1-lamu2] + [Atv; zeros(N,1)];
+    #  resnorm = norm([rdual; rcent; rpri]);
+    #  
+    #  done = (sdg < pdtol) | (pditer >= pdmaxiter);
+    #  
+    #  disp(sprintf('Iteration = #d, tau = #8.3e, Primal = #8.3e, PDGap = #8.3e, Dual res = #8.3e, Primal res = #8.3e',...
+    #    pditer, tau, sum(u), sdg, norm(rdual), norm(rpri)));
+    #  if (largescale)
+    #    disp(sprintf('                  CG Res = #8.3e, CG Iter = #d', cgres, cgiter));
+    #  else
+    #    disp(sprintf('                  H11p condition number = #8.3e', hcond));
+    #  end
+    #  
+    #end
+    
+    # End of original Matab code
+    #----------------------------    
+    
+    #largescale = isa(A,'function_handle');
+    if hasattr(A, '__call__'):
+        largescale = True
+    else:
+        largescale = False       
+    
+    #N = length(x0);
+    N = x0.size
+    
+    alpha = 0.01
+    beta = 0.5
+    mu = 10
+    
+    #gradf0 = [zeros(N,1); ones(N,1)];
+    gradf0 = numpy.hstack((numpy.zeros(N), numpy.ones(N)))
+    
+    # starting point --- make sure that it is feasible
+    #if (largescale)
+    if largescale:
+      raise l1qcNotImplementedError('Largescale not implemented yet!')
+    else:
+      #if (norm(A*x0-b)/norm(b) > cgtol)
+      if numpy.linalg.norm(numpy.dot(A,x0)-b) / numpy.linalg.norm(b) > cgtol:
+        #disp('Starting point infeasible; using x0 = At*inv(AAt)*y.');
+        if verbose:
+          print '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;
+        try:
+            w = scipy.linalg.solve(numpy.dot(A,A.T), b, sym_pos=True)
+            hcond = 1.0/numpy.linalg.cond(np.dot(A,A.T))
+        except scipy.linalg.LinAlgError:
+            if verbose:
+              print 'A*At is ill-conditioned: cannot find starting point'
+            xp = x0.copy()
+            return xp
+        if hcond < 1e-14:
+            if verbose:
+              print 'A*At is ill-conditioned: cannot find starting point'
+            xp = x0.copy()
+            return xp           
+        x0 = numpy.dot(A.T, w)
+      #end  
+    #end
+    x = x0.copy()
+    #u = (0.95)*abs(x0) + (0.10)*max(abs(x0));
+    u = (0.95)*numpy.abs(x0) + (0.10)*numpy.abs(x0).max()
+    
+    # set up for the first iteration
+    fu1 = x - u
+    fu2 = -x - u
+    lamu1 = -1/fu1
+    lamu2 = -1/fu2
+    if (largescale):
+      #v = -A(lamu1-lamu2);
+      #Atv = At(v);
+      #rpri = A(x) - b;
+      raise l1qcNotImplementedError('Largescale not implemented yet!')
+    else:
+      #v = -A*(lamu1-lamu2);
+      #Atv = A'*v;
+      #rpri = A*x - b;
+      v = numpy.dot(-A, lamu1-lamu2)
+      Atv = numpy.dot(A.T, v)
+      rpri = numpy.dot(A,x) - b
+    #end
+    
+    #sdg = -(fu1'*lamu1 + fu2'*lamu2);
+    sdg = -(numpy.dot(fu1,lamu1) + numpy.dot(fu2,lamu2))
+    tau = mu*2*N/sdg
+    
+    #rcent = [-lamu1.*fu1; -lamu2.*fu2] - (1/tau);
+    rcent = numpy.hstack((-numpy.dot(lamu1,fu1),  -numpy.dot(lamu2,fu2))) - (1/tau)
+    #rdual = gradf0 + [lamu1-lamu2; -lamu1-lamu2] + [Atv; zeros(N,1)];
+    rdual = gradf0 + numpy.hstack((lamu1-lamu2, -lamu1-lamu2)) + numpy.hstack((Atv, numpy.zeros(N)))
+    #resnorm = norm([rdual; rcent; rpri]);
+    resnorm = numpy.linalg.norm(numpy.hstack((rdual, rcent, rpri)))
+    
+    pditer = 0
+    #done = (sdg < pdtol) | (pditer >= pdmaxiter);
+    done = (sdg < pdtol) or (pditer >= pdmaxiter)
+    #while (~done)
+    while not done:
+      
+      pditer = pditer + 1
+      
+      #w1 = -1/tau*(-1./fu1 + 1./fu2) - Atv;
+      w1 = -1/tau*(-1/fu1 + 1/fu2) - Atv
+      w2 = -1 - 1/tau*(1/fu1 + 1/fu2)
+      w3 = -rpri
+      
+      #sig1 = -lamu1./fu1 - lamu2./fu2;
+      sig1 = -lamu1/fu1 - lamu2/fu2
+      sig2 = lamu1/fu1 - lamu2/fu2
+      #sigx = sig1 - sig2.^2./sig1;
+      sigx = sig1 - sig2**2/sig1
+      
+      if largescale:
+        #w1p = w3 - A(w1./sigx - w2.*sig2./(sigx.*sig1));
+        #h11pfun = @(z) -A(1./sigx.*At(z));
+        #[dv, 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
+        #dx = (w1 - w2.*sig2./sig1 - At(dv))./sigx;
+        #Adx = A(dx);
+        #Atdv = At(dv);
+        raise l1qcNotImplementedError('Largescale not implemented yet!')
+      else:
+        #w1p = -(w3 - A*(w1./sigx - w2.*sig2./(sigx.*sig1)));
+        w1p = -(w3 - numpy.dot(A,(w1/sigx - w2*sig2/(sigx*sig1))))
+        #H11p = A*(sparse(diag(1./sigx))*A');
+        H11p = numpy.dot(A, numpy.dot(numpy.diag(1/sigx),A.T))
+        #opts.POSDEF = true; opts.SYM = true;
+        #[dv,hcond] = linsolve(H11p, w1p, opts);
+        try:
+          dv = scipy.linalg.solve(H11p, w1p, sym_pos=True)
+          hcond = 1.0/numpy.linalg.cond(H11p)
+        except scipy.linalg.LinAlgError:
+            if verbose:
+              print 'Matrix ill-conditioned.  Returning previous iterate.  (See Section 4 of notes for more information.)'
+            xp = x.copy()
+            return xp
+        if hcond < 1e-14:
+            if verbose:
+              print 'Matrix ill-conditioned.  Returning previous iterate.  (See Section 4 of notes for more information.)'
+            xp = x.copy()
+            return xp            
+        #if (hcond < 1e-14)
+        #  disp('Matrix ill-conditioned.  Returning previous iterate.  (See Section 4 of notes for more information.)');
+        #  xp = x;
+        #  return
+        #end
+        
+        #dx = (w1 - w2.*sig2./sig1 - A'*dv)./sigx;
+        dx = (w1 - w2*sig2/sig1 - numpy.dot(A.T,dv))/sigx
+        #Adx = A*dx;
+        Adx = numpy.dot(A,dx)
+        #Atdv = A'*dv;
+        Atdv = numpy.dot(A.T,dv)
+      #end
+      
+      #du = (w2 - sig2.*dx)./sig1;
+      du = (w2 - sig2*dx)/sig1
+      
+      #dlamu1 = (lamu1./fu1).*(-dx+du) - lamu1 - (1/tau)*1./fu1;
+      dlamu1 = (lamu1/fu1)*(-dx+du) - lamu1 - (1/tau)*1/fu1
+      dlamu2 = (lamu2/fu2)*(dx+du) - lamu2 - 1/tau*1/fu2
+      
+      # make sure that the step is feasible: keeps lamu1,lamu2 > 0, fu1,fu2 < 0
+      #indp = find(dlamu1 < 0);  indn = find(dlamu2 < 0);
+      indp = numpy.nonzero(dlamu1 < 0)
+      indn = numpy.nonzero(dlamu2 < 0)
+      #s = min([1; -lamu1(indp)./dlamu1(indp); -lamu2(indn)./dlamu2(indn)]);
+      s = numpy.min(numpy.hstack((numpy.array([1]), -lamu1[indp]/dlamu1[indp], -lamu2[indn]/dlamu2[indn])))
+      #indp = find((dx-du) > 0);  indn = find((-dx-du) > 0);
+      indp = numpy.nonzero((dx-du) > 0)
+      indn = numpy.nonzero((-dx-du) > 0)
+      #s = (0.99)*min([s; -fu1(indp)./(dx(indp)-du(indp)); -fu2(indn)./(-dx(indn)-du(indn))]);
+      s = (0.99)*numpy.min(numpy.hstack((numpy.array([s]), -fu1[indp]/(dx[indp]-du[indp]), -fu2[indn]/(-dx[indn]-du[indn]))))
+      
+      # backtracking line search
+      suffdec = 0
+      backiter = 0
+      #while (~suffdec)
+      while not suffdec:
+        #xp = x + s*dx;  up = u + s*du; 
+        xp = x + s*dx
+        up = u + s*du 
+        #vp = v + s*dv;  Atvp = Atv + s*Atdv; 
+        vp = v + s*dv
+        Atvp = Atv + s*Atdv
+        #lamu1p = lamu1 + s*dlamu1;  lamu2p = lamu2 + s*dlamu2;
+        lamu1p = lamu1 + s*dlamu1
+        lamu2p = lamu2 + s*dlamu2
+        #fu1p = xp - up;  fu2p = -xp - up;  
+        fu1p = xp - up
+        fu2p = -xp - up
+        #rdp = gradf0 + [lamu1p-lamu2p; -lamu1p-lamu2p] + [Atvp; zeros(N,1)];
+        rdp = gradf0 + numpy.hstack((lamu1p-lamu2p, -lamu1p-lamu2p)) + numpy.hstack((Atvp, numpy.zeros(N)))
+        #rcp = [-lamu1p.*fu1p; -lamu2p.*fu2p] - (1/tau);
+        rcp = numpy.hstack((-lamu1p*fu1p, -lamu2p*fu2p)) - (1/tau)
+        #rpp = rpri + s*Adx;
+        rpp = rpri + s*Adx
+        #suffdec = (norm([rdp; rcp; rpp]) <= (1-alpha*s)*resnorm);
+        suffdec = (numpy.linalg.norm(numpy.hstack((rdp, rcp, rpp))) <= (1-alpha*s)*resnorm)
+        s = beta*s
+        backiter = backiter + 1
+        if (backiter > 32):
+          if verbose:
+            print 'Stuck backtracking, returning last iterate.  (See Section 4 of notes for more information.)'
+          xp = x.copy()
+          return xp
+        #end
+      #end
+      
+      
+      # next iteration
+      #x = xp;  u = up;
+      x = xp.copy()
+      u = up.copy()
+      #v = vp;  Atv = Atvp; 
+      v = vp.copy()
+      Atv = Atvp.copy()
+      #lamu1 = lamu1p;  lamu2 = lamu2p;
+      lamu1 = lamu1p.copy()
+      lamu2 = lamu2p.copy()
+      #fu1 = fu1p;  fu2 = fu2p;
+      fu1 = fu1p.copy()
+      fu2 = fu2p.copy()
+      
+      # surrogate duality gap
+      #sdg = -(fu1'*lamu1 + fu2'*lamu2);
+      sdg = -(numpy.dot(fu1,lamu1) + numpy.dot(fu2,lamu2))
+      tau = mu*2*N/sdg
+      rpri = rpp.copy()
+      #rcent = [-lamu1.*fu1; -lamu2.*fu2] - (1/tau);
+      rcent = numpy.hstack((-lamu1*fu1, -lamu2*fu2)) - (1/tau)
+      #rdual = gradf0 + [lamu1-lamu2; -lamu1-lamu2] + [Atv; zeros(N,1)];
+      rdual = gradf0 + numpy.hstack((lamu1-lamu2, -lamu1-lamu2)) + numpy.hstack((Atv, numpy.zeros(N)))
+      #resnorm = norm([rdual; rcent; rpri]);
+      resnorm = numpy.linalg.norm(numpy.hstack((rdual, rcent, rpri)))
+      
+      #done = (sdg < pdtol) | (pditer >= pdmaxiter);
+      done = (sdg < pdtol) or (pditer >= pdmaxiter)
+      
+      if verbose:
+        print 'Iteration =',pditer,', tau =',tau,', Primal =',numpy.sum(u),', PDGap =',sdg,', Dual res =',numpy.linalg.norm(rdual),', Primal res =',numpy.linalg.norm(rpri)
+      if largescale:
+        #disp(sprintf('                  CG Res = #8.3e, CG Iter = #d', cgres, cgiter));
+        raise l1qcNotImplementedError('Largescale not implemented yet!')
+      else:
+        #disp(sprintf('                  H11p condition number = #8.3e', hcond));
+        if verbose:
+          print '                  H11p condition number =',hcond
+      #end
+      
+    #end
+    
+    return xp