Codes for Paper Local Second-Order Limit Dynamics of the Alternating Direction Method of Multipliers for Semidefinite Programming

All codes are ready to run. No need to install anything except MATLAB.

1. $\sigma$ updating rules

The $\sigma$ updating rules for the first $20,000$ iterations can be found in admm-single-light/update_sig_1.m.

2. File execution order

Toy examples

Directly run toy-examples/toy1.m, toy-examples/toy2.m, and toy-examples/toy3.m separately. All results are already contained in the toy-examples/figs folder.

Mittelmann dataset

For any instances out of the selected $25$ Mittelmann problems (single block, matrix size no greater than $3,000$):

1. Run sig-wave/generate_start_points.m to generate starting points for both the three-step ADMM formal test and $\sigma$ tuning test.
2. After (1), run acceleration/main_baseline.m to generate the three-step ADMM long trajectories $\lVert \Delta Z^{(k)} \rVert_F$, $r_{\max}^{(k)}$, and $\angle(\Delta Z^{(k)}, \Delta Z^{(k+1)})$. The maximum iteration number is set to $1,000,000$.
3. After (1), we can also directly run sig-wave/study_sig_wave.m to generate $\sigma$ tuning trajectories $r_p^{(k)}$, $r_{d}^{(k)}$, $\lVert \Delta X^{(k)} \rVert_F$, and $\lVert \Delta S^{(k)} \rVert_F$.

Please change the type to the data instance you want to test in each folder. After running all $25$ instances, you can then use sig-wave/plot_sig_wave.m and acceleration/plot_all.m to generate figures in the paper. The results are also already contained in the acceleration/figs folder.