Here’s one way to do it. I modified proj_psdUTrace.m to take in an extra trace term (since this is just a linear term). An extra linear term is quite common so we may take care of it nicely in a future release of TFOCS. For now, the modified proj_psdUTrace.m is in github, so head there to get the latest version.
First, setup the problem
rng(100);
d = 20;
X = randn(d);
R = X'*X; % Sigma
e = ones(d,1);
lambda = 0.5;
Now solve in CVX so we can verify we’re solving the right problem
cvx_begin
cvx_precision best
variable X(d,d) semidefinite
minimize( -R(:)'*X(:)+lambda*norm(X(:),1) )
subject to
trace(X)==1;
cvx_end
toc
X_CVX = X;
And the TFOCS solve:
mu = 1e-2;
largescale = ( d > 200 );
is_real = true;
obj= proj_psdUTrace( 1, largescale, is_real, [], -R );
linear = linop_vec( [d,d] );
affine = { linear };
dualProx = proj_linf(lambda);
x0 = [];
z0 = [];
opts = struct('printEvery',4,'tol',1e-10,'maxIts',500,'L0',1);
opts.errFcn = @(f,d,p) norm(p-X_CVX,'fro')/norm(X_CVX,'fro');
opts.restart = 50;
[x,out,optsOut] = tfocs_SCD( obj, affine, dualProx, mu, x0, z0, opts );
which gives the following output on my computer:
Auslender & Teboulle's single-projection method
Iter Objective |dx|/|x| step errors
----+----------------------------------+----------
4 | -6.82076e+01 2.53e-01 1.52e+00 | 1.06e-01
8 | -6.70352e+01 9.12e-02 2.32e+00 | 6.26e-02
12 | -6.67141e+01 4.09e-02 3.54e+00 | 2.97e-02
16 | -6.66041e+01 2.35e-02 5.40e+00 | 1.44e-02
20 | -6.65586e+01 1.25e-02 8.23e+00 | 7.41e-03
24 | -6.65371e+01 7.68e-03 1.25e+01 | 4.15e-03
28 | -6.65259e+01 5.42e-03 1.91e+01 | 2.42e-03
32 | -6.65197e+01 3.74e-03 2.91e+01 | 1.43e-03
...
100 | -6.65105e+01 2.10e-06 2.63e+01*| 2.39e-07
101 | -6.65105e+01 4.26e-05 2.93e+01 | 3.13e-08 | restarted
104 | -6.65105e+01 3.68e-08 9.84e+00*| 1.11e-08
108 | -6.65105e+01 6.13e-09 1.50e+01*| 1.97e-09
112 | -6.65105e+01 2.03e-09 1.04e+01*| 3.20e-10
116 | -6.65105e+01 5.92e-10 1.59e+01*| 2.86e-10
120 | -6.65105e+01 1.24e-10 2.42e+01*| 1.72e-10
123 | -6.65105e+01 9.59e-11 1.32e+01*| 1.71e-10
Finished: Step size tolerance reached