Sleipnir

Sparsity and Linearity-Exploiting Interior-Point solver - Now Internally Readable

Named after Odin's eight-legged horse from Norse mythology, Sleipnir is a linearity-exploiting sparse nonlinear constrained optimization problem solver that uses the interior-point method.

```cpp

include

include

int main() { // Find the x, y pair with the largest product for which x + 3y = 36 slp::Problem problem;

auto x = problem.decisionvariable(); auto y = problem.decisionvariable();

problem.maximize(x * y); problem.subject_to(x + 3 * y == 36); problem.solve();

// x = 18.0, y = 6.0 std::println("x = {}, y = {}", x.value(), y.value()); } ```

```python

!/usr/bin/env python3

from jormungandr.optimization import Problem

def main(): # Find the x, y pair with the largest product for which x + 3y = 36 problem = Problem()

x = problem.decision_variable()
y = problem.decision_variable()

problem.maximize(x * y)
problem.subject_to(x + 3 * y == 36)
problem.solve()

# x = 18.0, y = 6.0
print(f"x = {x.value()}, y = {y.value()}")

if name == "main": main() ```

Sleipnir's internals are intended to be readable by those who aren't domain experts with links to explanatory material for its algorithms.

Benchmarks

flywheel-scalability-results-casadi.csv flywheel-scalability-results-sleipnir.csv	cart-pole-scalability-results-casadi.csv cart-pole-scalability-results-sleipnir.csv

Generated by tools/generate-scalability-results.sh from benchmarks/scalability source.

• CPU: AMD Ryzen 7 7840U
• RAM: 64 GB, 5600 MHz DDR5
• Compiler version: g++ (GCC) 15.1.1 20250425

The following thirdparty software was used in the benchmarks:

• CasADi 3.7.0 (autodiff and NLP solver frontend)
• Ipopt 3.14.17 (NLP solver backend)
• MUMPS 5.7.3 (linear solver)

Ipopt uses MUMPS by default because it has free licensing. Commercial linear solvers may be much faster.

See benchmark details for more.

Install

Minimum system requirements

• Windows
  • OS: Windows 10
  • Runtime: Microsoft Visual C++ 2022 redistributable from Visual Studio 2022 17.13
• Linux
  • OS: Ubuntu 24.04
  • Runtime: GCC 14 libstdc++ (run sudo apt install g++-14)
• macOS
  • OS: macOS 14.5
  • Runtime: Apple Clang 16.0.0 libc++ from Xcode 16.2 (run xcode-select --install)

C++ library

To install Sleipnir system-wide, see the build instructions.

To use Sleipnir within a CMake project, add the following to your CMakeLists.txt: ```cmake include(FetchContent)

fetchcontentdeclare( Sleipnir GITREPOSITORY https://github.com/SleipnirGroup/Sleipnir GITTAG main EXCLUDEFROMALL SYSTEM ) fetchcontentmakeavailable(Sleipnir)

targetlinklibraries(MyApp PUBLIC Sleipnir::Sleipnir) ```

Python library

bash pip install sleipnirgroup-jormungandr

API docs

See the C++ API docs and Python API docs.

Examples

See the examples, C++ optimization unit tests, and Python optimization unit tests.

Build

Dependencies

• C++23 compiler
  • On Windows 10 or greater, install Visual Studio Community 2022 and select the C++ programming language during installation
  • On Ubuntu 24.04 or greater, install GCC 14 via sudo apt install g++-14
  • On macOS 14.5 or greater, install the Xcode 16.2 command-line build tools via xcode-select --install
• CMake 3.21 or greater
  • On Windows, install from the link above
  • On Linux, install via sudo apt install cmake
  • On macOS, install via brew install cmake
• Python 3.10 or greater
  • On Windows, install from the link above
  • On Linux, install via sudo apt install python
  • On macOS, install via brew install python
• Eigen
• small_vector
• nanobind (build only)
• Catch2 (tests only)

Library dependencies which aren't installed locally will be automatically downloaded and built by CMake.

The benchmark executables require CasADi to be installed locally.

C++ library

On Windows, open a Developer PowerShell. On Linux or macOS, open a Bash shell.

```bash

Clone the repository

git clone git@github.com:SleipnirGroup/Sleipnir cd Sleipnir

Configure; automatically downloads library dependencies

cmake -B build -S .

Build

cmake --build build

Test

ctest --test-dir build --output-on-failure

Install

cmake --install build --prefix pkgdir ```

The following build types can be specified via -DCMAKE_BUILD_TYPE during CMake configure:

• Debug
  • Optimizations off
  • Debug symbols on
• Release
  • Optimizations on
  • Debug symbols off
• RelWithDebInfo (default)
  • Release build type, but with debug info
• MinSizeRel
  • Minimum size release build
• Asan
  • Enables address sanitizer
• Tsan
  • Enables thread sanitizer
• Ubsan
  • Enables undefined behavior sanitizer
• Perf
  • RelWithDebInfo build type, but with frame pointer so perf utility can use it

Python library

On Windows, open a Developer PowerShell. On Linux or macOS, open a Bash shell.

```bash

Clone the repository

git clone git@github.com:SleipnirGroup/Sleipnir cd Sleipnir

Setup

pip install --user build

Build

python -m build --wheel

Install

pip install --user dist/sleipnirgroup_jormungandr-*.whl

Test

pytest ```

Test diagnostics

Passing the --enable-diagnostics flag to the test executable enables solver diagnostic prints.

Some test problems generate CSV files containing their solutions. These can be plotted with tools/plottestproblem_solutions.py.

Benchmark details

Running the benchmarks

Benchmark projects are in the benchmarks folder. To compile and run them, run the following in the repository root: ```bash

Install CasADi and [matplotlib, numpy, scipy] pip packages first

cmake -B build -S . -DBUILD_BENCHMARKS=ON cmake --build build ./tools/generate-scalability-results.sh ```

See the contents of ./tools/generate-scalability-results.sh for how to run specific benchmarks.

How we improved performance

Make more decisions at compile time

During problem setup, equality and inequality constraints are encoded as different types, so the appropriate setup behavior can be selected at compile time via operator overloads.

Reuse autodiff computation results that are still valid (aka caching)

The autodiff library automatically records the linearity of every node in the computational graph. Linear functions have constant first derivatives, and quadratic functions have constant second derivatives. The constant derivatives are computed in the initialization phase and reused for all solver iterations. Only nonlinear parts of the computational graph are recomputed during each solver iteration.

For quadratic problems, we compute the Lagrangian Hessian and constraint Jacobians once with no problem structure hints from the user.

Use a performant linear algebra library with fast sparse solvers

Eigen provides these. It also has no required dependencies, which makes cross compilation much easier.

Use a pool allocator for autodiff expression nodes

This promotes fast allocation/deallocation and good memory locality.

We could mitigate the solver's high last-level-cache miss rate (~42% on the machine above) further by breaking apart the expression nodes into fields that are commonly iterated together. We used to use a tape, which gave computational graph updates linear access patterns, but tapes are monotonic buffers with no way to reclaim storage.