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Diffstat (limited to 'Optimization/l1decode_pd.m')
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diff --git a/Optimization/l1decode_pd.m b/Optimization/l1decode_pd.m new file mode 100644 index 0000000..2757404 --- /dev/null +++ b/Optimization/l1decode_pd.m @@ -0,0 +1,175 @@ +% 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 + |