# On the Adaptive Penalty Parameter Selection in ADMM
## 3 Numerical Experiments
### Portfolio Optimization
The minimization with respect to \(v\) can be carried out efficiently using the soft thresholding operator. We test the effectiveness on the real market data. We show results obtained using the following datasets [26]:  
* FF48 (Fama & French 48 Industry portfolios): Contain monthly returns of 48 from July 1926 to December 2015. We simulate investment strategies of length 10, 20, and 30 years, with annual rebalancing. * NASDAQ100 (NASDAQ 100 stock Price Data): Contains monthly returns from November 2004 to April 2016. We simulate investment strategy of length 10, with annual rebalancing. * Dow Jones (Dow Jones Industrial): Contains monthly returns from February 1990 to April 2016. We simulate investment strategy of length 10, with annual rebalancing. We set for all the tests \(\lambda=10^{-2}\) to enforce a sparse portfolio. We compared the 5 considered algorithms in terms of number of iterations needed to reach a tolerance \(10^{-5}\) on the relative residual, letting them run for a maximum of 3000 iterations. Moreover, we considered some financial performance measures expressing the goodness of the optimal portfolios, i.e.,  
* density: the number of nonzero elements in the solution (percentage) that gives an estimation of holding cost;
* ratio: it estimates the risk reduction when the naive strategy (at each rebalancing date the total wealth is equally divided among the assets) is taken as the benchmark, defined as \[ratio=\frac{u_{naive}^{\top}\mathbb{C}u_{naive}}{u_{*}^{\top}\mathbb{C}u_{*}},\] where the numerator is the variance of the portfolio produced by the naive strategy and the denominator is the variance of the optimal portfolio. Tables 2-6 report the results of the experiments performed on the 5 datasets with different choices for the value of \(\tau_{0}\) (namely, 0.1, 0.5, 1) for a total of 15 instances. The five algorithms are overall able to obtain equivalent portfolios when converging to the desired tolerance. However, they behave quite differently in terms of iterations needed to converge. In general, the adaptive strategies allow a reduction in the computational complexity with respect to the Vanilla ADMM, which is unable to reach the desired tolerance in 9 out of 15 instances. Among the 4 adaptive strategies, \"Adaptive ADMM-BB1\" seems to be the most effective, being able to outperform all the others in 9 out of 15 instances, and being the second best in 4 out of 15, performing on average 20% of iterations more than the best in this case. Unlike what happened in the case of consensus logistic regression problems, here \"Adaptive ADMM-BB2\" and \"Adaptive ADMM-ABB\({}_{\min}\)\" appear to perform poorly, suggesting that the use of too small of values for \(\tau^{k}\) may slow down the convergence. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Method** & **Iterations** & **Density** & **Ratio** \\ \hline \multicolumn{4}{c}{\(\tau^{0}=0.1\)} \\ Vanilla ADMM & 846 & 0.096 & 2.064 \\ Adaptive ADMM [7] & 321 & 0.096 & 2.064 \\ Adaptive ADMM-BB1 & 190 & 0.096 & 2.064 \\ Adaptive ADMM-BB2 & 458 & 0.096 & 2.064 \\ Adaptive ADMM-ABB\({}_{\min}\) & 509 & 0.096 & 2.064 \\  & & \(\tau^{0}=0.5\) & \\ Vanilla ADMM & 3000 & — & — \\ Adaptive ADMM [7] & 488 & 0.096 & 2.064 \\ Adaptive ADMM-BB1 & 340 & 0.096 & 2.064 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance for portfolio FF48 with 10-year simulation.