This Matlab function contains our ADMM implementation with the sharing scheme to compute proximal operator of the LOG penalty. For more informatio, please refer to Zhang, D., Liu, Y., Davanloo Tajbakhsh, S. (2020), A first-order optimization algorithm for statistical learning with hierarchical sparsity structure

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+ --- prox_ADMM.m : solver for proximal operator of LOG penalty using ADMM
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+ --- driver.m : Demo code for calling prox_ADMM
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+ --- Compare_Algrhm : Code that compare different method of proximal operator

Function:

[beta,objval,penaltyval] = prox_ADMM(ancestor,b,iter_max,rho,lambda,w);

solves problem $$ \min_{\beta} \frac{1}{2}||\beta-b||^2 + \lambda \Omega_{\text{LOG}}(\beta) $$ Required input:

• ancestor: cell array that stores the ancestor nodes of each node
• b: vector
• iter_max: maximum number of iteration
• rho : stepsize of ADMM
• lambda: parameter > 0
• w : penalty value $\omega$ for each group

Output:

• beta : solution of proximal operator
• objval : objective value of proximal operator
• penaltyval : value of LOG penalty $ \lambda \Omega_{\text{LOG}}(\beta) $

Another way to set ancestor is provided in driver.m. Users can define an $n\times 2$ matrix 'indx_arc' with each row to be a directed edge of DAG. Then the code below will generate the cell array ancestor.

%% Generate index of ancestor for each node
ancestor = cell(n_DAG,1);
for i = 1:n_DAG
    idx_ancestor = indx_arc(:, 2) == i;
    ancestor{i} = indx_arc(idx_ancestor, 1);
    ancestor{i} = unique([i;cell2mat({ancestor{[indx_arc(idx_ancestor, 1); i]}}')]);
end