QP solvers benchmark

Benchmark for quadratic programming (QP) solvers available in Python.

The objective is to compare and select the best QP solvers for given use cases. The benchmarking methodology is open to discussions. Standard and community test sets are available: all of them can be processed using the qpbenchmark command-line tool, resulting in standardized reports evaluating all metrics across all QP solvers available on the test machine.

Test sets

The benchmark comes with standard and community test sets to represent different use cases for QP solvers:

• Free-for-all: community-built test set, new problems welcome!
• Maros-Meszaros: a standard test set with problems designed to be difficult.
• Model predictive control: model predictive control problems arising e.g. in robotics.

New test sets are welcome! The qpbenchmark tool is designed to make it easy to wrap up a new test set without re-implementing the benchmark methodology. Check out the contribution guidelines to get started.

Solvers

Solver	Keyword	Algorithm	Matrices	License
Clarabel	clarabel	Interior point	Sparse	Apache-2.0
CVXOPT	cvxopt	Interior point	Dense	GPL-3.0
DAQP	daqp	Active set	Dense	MIT
ECOS	ecos	Interior point	Sparse	GPL-3.0
Gurobi	gurobi	Interior point	Sparse	Commercial
HiGHS	highs	Active set	Sparse	MIT
HPIPM	hpipm	Interior point	Dense	BSD-2-Clause
MOSEK	mosek	Interior point	Sparse	Commercial
NPPro	nppro	Active set	Dense	Commercial
OSQP	osqp	Douglas–Rachford	Sparse	Apache-2.0
PIQP	piqp	Proximal Interior Point	Dense & Sparse	BSD-2-Clause
ProxQP	proxqp	Augmented Lagrangian	Dense & Sparse	BSD-2-Clause
QPALM	qpalm	Augmented Lagrangian	Sparse	LGPL-3.0
qpOASES	qpoases	Active set	Dense	LGPL-2.1
qpSWIFT	qpswift	Interior point	Sparse	GPL-3.0
quadprog	quadprog	Goldfarb-Idnani	Dense	GPL-2.0
SCS	scs	Douglas–Rachford	Sparse	MIT

Metrics

We evaluate QP solvers based on the following metrics:

• Success rate: percentage of problems a solver is able to solve on a given test set.
• Computation time: time a solver takes to solve a given problem.
• Optimality conditions: we evaluate all three optimality conditions:
  • Primal residual: maximum error on equality and inequality constraints at the returned solution.
  • Dual residual: maximum error on the dual feasibility condition at the returned solution.
  • Duality gap: value of the duality gap at the returned solution.

Shifted geometric mean

Each metric (computation time, primal and dual residuals, duality gap) produces a different ranking of solvers for each problem. To aggregate those rankings into a single metric over the whole test set, we use the shifted geometric mean (shm), which is a standard to aggregate computation times in benchmarks for optimization software. This mean has the advantage of being compromised by neither large outliers (as opposed to the arithmetic mean) nor by small outliers (in contrast to the geometric geometric mean). Check out the references below for further details.

Intuitively, a solver with a shifted-geometric-mean runtime of $Y$ is $Y$ times slower than the best solver over the test set. Similarly, a solver with a shifted-geometric-mean primal residual $R$ is $R$ times less accurate on equality and inequality constraints than the best solver over the test set.

Results

The outcome from running a test set is a standardized report comparing solvers against the different metrics. Here are the results for the various qpbenchmark test sets:

• Free-for-all results
• Maros-Meszaros results
• Model predictive control results

You can check out results from a variety of machines, and share the reports produced by running the benchmark on your own machine, in the Results category of the discussions forum of each test set.

Limitations

Here are some known areas of improvement for this benchmark:

• Cold start only: we don't evaluate warm-start performance for now.
• CPU thermal throttling: the benchmark currently does not check the status of CPU thermal throttling.
  • Adding this feature is a good way to start contributing to the benchmark.
• HPIPM evaluation: currently we evaluate HPIPM against a duality-gap tolerance but the solver is asked to solve a complementarity-slackness condition.

Check out the issue tracker for ongoing works and future improvements.

Installation

We recommend installing the benchmark and all solvers in an isolated environment using conda:

conda env create -f environment.yaml
conda activate qpbenchmark

Alternatively, you can install the benchmarking tool individually by pip install qpbenchmark. In that case, the benchmark will run on all supported solvers it can import.

Usage

The benchmark works by running qpbenchmark on a Python script describing the test set. For instance:

qpbenchmark my_test_set.py run

The test-set script is followed by a benchmark command, such as "run" here. We can add optional arguments to run a specific solver, problem, or solver settings:

qpbenchmark my_test_set.py run --solver proxqp --settings default

Check out qpbenchmark --help for a list of available commands and arguments.

Plots

The command line ships a plot command to compare solver performances over a test set for a specific metric. For instance, run:

qpbenchmark maros_meszaros_dense.py plot runtime high_accuracy

To generate the following plot:

Contributing

Contributions to improving this benchmark are welcome. You can for instance propose new problems, or share the runtimes you obtain on your machine. Check out the contribution guidelines for details.

Citation

If you use qpbenchmark in your works, please cite all its contributors as follows:

@software{qpbenchmark2024,
  title = {{qpbenchmark: Benchmark for quadratic programming solvers available in Python}},
  author = {Caron, Stéphane and Zaki, Akram and Otta, Pavel and Arnström, Daniel and Carpentier, Justin and Yang, Fengyu and Leziart, Pierre-Alexandre},
  url = {https://github.com/qpsolvers/qpbenchmark},
  license = {Apache-2.0},
  version = {2.3.0},
  year = {2024}
}

See also

References

• How not to lie with statistics: the correct way to summarize benchmark results: why geometric means should always be used to summarize normalized results.
• Optimality conditions and numerical tolerances in QP solvers: note written while figuring out the high_accuracy settings of this benchmark.

Other benchmarks

• BenchOpt: a benchmarking suite tailored for machine learning workflows.
• Benchmarks for optimization software by Hans Mittelmann, which includes reports on the Maros-Meszaros test set.
• jrl-qp/benchmarks: benchmark of QP solvers available in C++.
• osqp_benchmarks: benchmark examples for the OSQP solver.
• proxqp_benchmark: benchmark examples for the ProxQP solver.
• qpmad_benchmark: benchmark examples for the qpmad solver.