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Dolan-Moré performance profiles

In order to compare different solvers, we employ the Dolan-Moré performance profile plot.

Let us briefly introduce the Dolan-Moré performance profile plot.

Let \(P\) be a finite set of problems used as benchmarks and \(S\) be a set of solvers we want to compare to one another.

Let \(t_{p,s}\) be the cost (e.g., runtime or flops) to solve a problem \(p\) using a solver \(s\).

We define the ration between \(t_{p,s}\) and the lowest observed cost to solve this problem using some solver \(s\in S\):

\begin{eqnarray*} r_{p,s} = \frac{t_{p,s}}{\min_{s \in S} t_{p,s}}. \end{eqnarray*}

If a solver \(s\) does not solve a problem \(p\), then we assign to \(r_{p,s}\) a very high value \(r_M > r_{p,s}\) for all other \(p,s\).

The cumulative distribution of the performance ratio is the Dolan-Moré performance profile plot.

In particular, define

\begin{eqnarray*} \rho_s(\tau) = \frac{1}{n_p}\#\{p\in P: r_{p,s}\leq \tau\}, \end{eqnarray*}

for \(\tau\geq 1\) and where \(n_p\) is the number of problems.

The Dolan-Moré performance profile is the plot of \(\rho_s\) vs \(\tau\), typically on a logarithmic x-axis.

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Benchmark results

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Benchmarking parameters

In all benchmark results presented below we set the tolerance to \(10^{-4}\).

The maximum number of iterations was set to a very high value above which we may confidently tell the problem is unlikely to be solved (e.g., \(10^6\)).

Given that different algorithms (SCS, SuperSCS using Broyden directions and SuperSCS using Anderson's acceleration) have a different per-iteration cost, we allow every algorithm to run for a give time (see max_time_milliseconds).

After that maximum time has passed, if the algorithm has not converged we consider that it has failed to solve the problem.

In Broyden's method we deactivated the K0 steps.

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LASSO problems

1152 lasso problems

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Regularized PCA

288 regularized PCA problems

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Logistic regression problems

288 logistic regression problems

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Semidefinite programming

48 SDP problems

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Ill-conditioned SDPs

48 ill-conditioned SDP problems

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Norm-constrained norm minimization

256 norm-constrained problems

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Maros-Meszaros Problems

We tested SuperSCS on the Maros-Meszaros collection of QP problems.

Find details here.