diff --git a/tex/sections/numerics.tex b/tex/sections/numerics.tex index b11fbec..7820671 100644 --- a/tex/sections/numerics.tex +++ b/tex/sections/numerics.tex @@ -33,7 +33,7 @@ \subsection{Implementation} \item {\tt HSL MA27/MA57}: Implement the \lblt factorization on the CPU~\cite{duff1983multifrontal}. It solves the augmented KKT system~\eqref{eq:kkt:augmented}. This solver serves as the reference when running on the CPU. - \item {\tt CHOLMOD}: Implements the Cholesky factorization on the CPU % Sungho: Maybe LDLFactorizations.jl instead, once the results are updated. + \item {\tt CHOLMOD}: Implements the Cholesky and \ldlt factorizations on the CPU % Sungho: Maybe LDLFactorizations.jl instead, once the results are updated. (using the AMD ordering \cite{amestoy-david-duff-2004} by default). It factorizes the condensed matrices $K_\gamma$ and $K_\tau$ appearing resp. in \eqref{eq:kkt:hykkt} and in \eqref{eq:liftedkkt}. @@ -186,7 +186,7 @@ \subsubsection{Tuning the equality relaxation strategy} We compare in Table~\ref{tab:sckkt:performance} the performance obtained by LiftedKKT as we decrease the IPM tolerance $\varepsilon_{tol}$. -We display both the runtimes on the CPU (using LDLFactorizations) and on the GPU (using {\tt cuDSS}-\ldlt). +We display both the runtimes on the CPU (using CHOLMOD-\ldlt) and on the GPU (using {\tt cuDSS}-\ldlt). The slacks associated with the relaxed equality constraints are converging to a value below $2 \tau$, leading to highly ill-conditioned terms in the diagonal matrices $\Sigma_s$. As a consequence, the conditioning of the matrix $K_\tau$ in \eqref{eq:liftedkkt} can increase @@ -200,24 +200,37 @@ \subsubsection{Tuning the equality relaxation strategy} \begin{table}[!ht] \centering \resizebox{.7\textwidth}{!}{ - \begin{tabular}{|l|rr|rr|r|} + % \begin{tabular}{|l|rr|rr|rr|r|} + % \hline + % & \multicolumn{2}{c|}{\bf CHOLMOD-\ldlt (CPU)} & \multicolumn{2}{c|}{\bf LDLFactorizations (CPU)} & \multicolumn{2}{c|}{\bf cuDSS-\ldlt (CUDA)}& \\ + % \hline + % $\varepsilon_{tol}$ & \#it & time (s)& \#it & time (s) & \#it & time (s) & accuracy \\ + % \hline + % $10^{-4}$& 115 & 268.2 &220& 358.8& 114 & 19.9& $1.2 \times 10^{-2}$\\ + % $10^{-5}$ & 210 & 777.8 & 120& 683.1& 113 & 30.4&$1.2 \times 10^{-3}$ \\ + % $10^{-6}$ & 102 & 337.5 & 109& 328.7& 109 & 25.0&$1.2 \times 10^{-4}$ \\ + % $10^{-7}$ &108 & 352.9 & 108& 272.9& 104 & 20.1&$1.2 \times 10^{-5}$ \\ + % $10^{-8}$ & - & - &- & - & 105 & 20.3&$1.2 \times 10^{-6}$ \\ + % \hline + % \end{tabular} + \begin{tabular}{|l|rr|rr|rr|} \hline - & \multicolumn{2}{c|}{\bf LDLFactorizations (CPU)} & \multicolumn{2}{c|}{\bf cuDSS-\ldlt (CUDA)}& \\ + & \multicolumn{2}{c|}{\bf CHOLMOD-\ldlt (CPU)} & \multicolumn{2}{c|}{\bf cuDSS-\ldlt (CUDA)}& \\ \hline - $\varepsilon_{tol}$ & \#it & time (s) & \#it & time (s) & accuracy \\ + $\varepsilon_{tol}$ & \#it & time (s)& \#it & time (s) & accuracy \\ \hline - $10^{-4}$ &220& 358.8& 114 & 19.9& $1.2 \times 10^{-2}$\\ - $10^{-5}$ &120& 683.1& 113 & 30.4&$1.2 \times 10^{-3}$ \\ - $10^{-6}$ &109& 328.7& 109 & 25.0&$1.2 \times 10^{-4}$ \\ - $10^{-7}$ &108& 272.9& 104 & 20.1&$1.2 \times 10^{-5}$ \\ - $10^{-8}$ & - & - & 105 & 20.3&$1.2 \times 10^{-6}$ \\ + $10^{-4}$ & 115 & 268.2 & 114 & 19.9 & $1.2 \times 10^{-2}$ \\ + $10^{-5}$ & 210 & 777.8 & 113 & 30.4 & $1.2 \times 10^{-3}$ \\ + $10^{-6}$ & 102 & 337.5 & 109 & 25.0 & $1.2 \times 10^{-4}$ \\ + $10^{-7}$ & 108 & 352.9 & 104 & 20.1 & $1.2 \times 10^{-5}$ \\ + $10^{-8}$ & - & - & 105 & 20.3 & $1.2 \times 10^{-6}$ \\ \hline \end{tabular} } \label{tab:sckkt:performance} \caption{Performance of the equality-relaxation strategy as we decrease the IPM tolerance $\varepsilon_{tol}$. - The table displays the wall time on the CPU (using LDLFactorizations) + The table displays the wall time on the CPU (using CHOLMOD-\ldlt) and on the GPU (using cuDSS-\ldlt). } \end{table}