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function [x,Z,z,ul,iters] = ...
bigM(F,blck_szs,c,x0,M,nu,abstol,reltol,tv,maxiters);
% [x,Z,z,ul,iters] = ...
% bigM(F,blck_szs,c,x0,M,nu,abstol,reltol,tv,maxiters);
%
% minimize c^T x
% subject to F(x) = F0 + x1*F1 + ... + xm*Fm >= 0
% Tr F(x) <= M
%
% maximize -Tr F0*(Z-zI) - Mz
% subject to Tr Fi*(Z-zI) = c_i
% Z >= 0, z>= 0
%
% Convergence criteria:
% (1) maxiters is exceeded
% (2) duality gap is less than abstol
% (3) primal and dual objective are both positive and
% duality gap is less than (reltol * dual objective)
% or primal and dual objective are both negative and
% duality gap is less than (reltol * minus the primal objective)
% (4) reltol is negative and
% primal objective is less than tv or dual objective is greater
% than tv
%
% Input arguments:
% F: (sum_i n_i^2) times (m+1) matrix
% [ F_0^1(:) F_1^1(:) ... F_m^1(:) ]
% [ F_0^2(:) F_1^2(:) ... F_m^2(:) ]
% ... ... ...
% [ F_0^L(:) F_1^L(:) ... F_m^L(:) ]
% F_i^j: jth block of F_i, size n_i times n_i.
% blck_szs: L-vector [n_1 ... n_L], dimensions of diagonal blocks.
% c: m-vector. Specifies primal objective.
% x0: m-vector. The primal starting point. F(x0) > 0.
% M: scalar. M > Tr F(x0).
% nu: >= 1.0. Controls the rate of convergence.
% abstol: absolute tolerance.
% reltol: relative tolerance. Has a special meaning when negative.
% tv: target value.
% maxiters: maximum number of iterations.
%
% Output arguments:
% x: m-vector; last primal iterate.
% Z: last dual iterate; block-diagonal matrix stored as
% [ Z^1(:); Z^2(:); ... ; Z^L(:) ].
% z: scalar part of last dual iterate.
% ul: ul(1): primal objective, ul(1): dual objective.
% iters: number of iterations taken.
if (size(blck_szs,2) < size(blck_szs,1)), blck_szs = blck_szs'; end;
if (size(blck_szs,1) ~= 1), error('blck_szs must be a vector'); end;
m = size(F,2)-1;
if (size(F,1) ~= sum(blck_szs.^2))
error('Dimensions of F do not match blck_szs.');
end;
if (size(x0,1) < size(x0,2)), x0 = x0'; end;
if (size(x0,1) ~= m) | (size(x0,2) ~= 1),
error('x0 must be an m-vector.');
end;
if (size(c,1) < size(c,2)), c = c'; end;
if (size(c,1) ~= m) | (size(c,2) ~= 1),
error('c must be an m-vector.');
end;
% I is the identity
I = zeros(size(F,1),1);
k=0; for n=blck_szs,
I(k+[1:n*n]) = reshape(eye(n),n*n,1); % identity
k = k+n*n; % k = sum n_i*n_i
end;
% Z0 = projection of I on dual feasible space
Z0 = I - F(:,2:m+1) * ...
( (F(:,2:m+1)'*F(:,2:m+1)) \ ( F(:,2:m+1)'*I - c ) );
% mineigZ is the smallest eigenvalue of Z0
mineigZ = 0.0;
k=0; for n=blck_szs,
mineigZ = min(mineigZ, min(eig(reshape(Z0(k+[1:n*n]),n,n))));
k=k+n*n;
end;
% z = max( 1e-5, -1.1*mineigZ )
Z0(k+1) = max( 1e-5, -1.1*mineigZ);
Z0(1:k) = Z0(1:k) + Z0(k+1)*I;
if (M < I'*F*[1;x0] + 1e-5),
error('M must be strictly greater than trace of F(x0).');
end;
% add scalar block Tr F(x) <= M
F = [F; M - I'*F(:,1), -I'*F(:,2:m+1)];
blck_szs = [blck_szs,1];
[x,Z,ul,infostr,time] = ...
sp(F,blck_szs,c,x0,Z0,nu,abstol,reltol,tv,maxiters);
z = Z(k+1);
Z = Z(1:k);
iters = time(3);
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