MC++: Toolkit for Construction, Manipulation and Evaluation of Factorable Functions

MC++ provides a collection of classes to support the construction and manipulation of factorable functions and their evaluation using a range of arithmetics. It is primarily written in C++ to promote execution speed, while the library PyMC++ provides Python binders through pybind11 (the module is imported as pymcpp).

Expression trees in MC++ can comprise any finite combination of unary and binary operations from a default library and are described by directed acyclic graphs (DAGs). MC++ also features a mechanism to extend DAGs with external operations, currently including affine and polynomial subexpressions, multi-layer perceptrons (MLP) and nested DAGs. Through the library CRONOS, MC++ can also be extended with systems of algebraic and differential equations. Expression trees generated with MC++ are used by the library CANON for local and global numerical optimization, and by the library MAGNUS for the development and analysis of mathematical models.

Capabilities

Version 5 of MC++ supports:

• Expression trees: construction of factorable functions as DAGs, with common-subexpression elimination; differentiation in both forward and reverse accumulation modes; evaluation in any of the arithmetics below; vectorized and multi-threaded evaluation over many points.
• Decomposition: recursive decomposition of factorable expressions into linear/polynomial subexpressions and transcendental operations, the basis for tailored relaxation and reformulation strategies.
• Nested expression trees: a DAG as an operation of another DAG, to enable tailored bounding strategies.
• External operations: user-defined operations with their own evaluation and differentiation rules -- the mechanism CRONOS uses to embed differential-equation solvers.

The bounding arithmetics, each a header that can be used on its own or through the DAGs:

arithmetic	purpose
Interval arithmetic	rigorous enclosures (Boost, PROFIL/BIAS or FILIB++ as backend)
Eigenvalue arithmetic	spectral bounds of Hessians
Ellipsoidal arithmetic	ellipsoidal enclosures of multivariate systems
McCormick relaxations	convex/concave relaxations and their subgradients
Taylor and Chebyshev models	polynomial models with rigorous remainder bounds
Polyhedral relaxations	linear relaxations for use in LP/MILP solvers
Superposition relaxations	relaxation via separable estimators

A range of Python scripts and notebooks in notebook/ illustrate these capabilities.

Setting up MC++

PyMC++ installs from PyPI, with pre-built wheels for Python 3.10 to 3.14 on Linux, macOS and Windows:

pip install pymcpp

To use the C++ headers, a different interval backend, or the HSL and Torch options, build from source with CMake; refer to INSTALL.md for the requirements, options and instructions.

Layout

src/mc/              the MC++ headers (header-only library)
src/3rdparty/        bundled third-party headers (FADBAD++, used only with MC__USE_FADBAD)
src/pymcpp/          the Python binders (PyMC++, module `pymcpp`)
test/                C++ test and example programs
notebook/            Python scripts and Jupyter notebooks

Contacts

• Repo owner: Benoit C. Chachuat
• OMEGA Research Group, Imperial College London

References

Methods implemented in MC++:

• Mitsos, A., B. Chachuat, P.I. Barton, McCormick-based relaxations of algorithms, SIAM Journal on Optimization, 20(2):573-601, 2009
• Bompadre, A., A. Mitsos, Convergence rate of McCormick relaxations, Journal of Global Optimization 52(1), 1-28, 2012
• Bompadre, A., A. Mitsos, B. Chachuat, Convergence analysis of Taylor models and McCormick-Taylor models, Journal of Global Optimization, 57(1), 75-114, 2013
• Tsoukalas, A., A. Mitsos, Multi-variate McCormick relaxations, Journal of Global Optimization, 59(2), 633-662, 2014
• Wechsung, A., P.I. Barton, Global optimization of bounded factorable functions with discontinuities, Journal of Global Optimization, 58(1), 1-30, 2014
• Villanueva, M.E., J. Rajyaguru, B. Houska, B. Chachuat, Ellipsoidal arithmetic for multivariate systems, Computer Aided Chemical Engineering, 37, 767-772, 2015
• Chachuat, B, B. Houska, R. Paulen, N. Peric, J. Rajyaguru, M.E. Villanueva, Set-theoretic approaches in analysis, estimation and control of nonlinear systems, IFAC-PapersOnLine, 48(8), 981-995, 2015
• Villanueva, M.E., Set-Theoretic Methods for Analysis, Estimation and Control of Nonlinear Systems, PhD Thesis, Department of Chemical Engineering, Imperial College London, 2016
• Rajyaguru, J., Villanueva M.E., Houska B., Chachuat B., Chebyshev model arithmetic for factorable functions, Journal of Global Optimization, 68, 413-438, 2017
• Karia, T., C.S. Adjiman, B. Chachuat, Assessment of a two-step approach for global optimization of mixed-integer polynomial programs using quadratic reformulation, Computers & Chemical Engineering, 165, 107909, 2022
• Zha, Y., M.E. Villanueva, B. Houska, B. Chachuat, Relaxation via separable estimators: Arithmetic and implementation, Journal of Global Optimization, in press, 2026

Software building on MC++:

• Bongartz, D., J. Najman, S. Sass, A. Mitsos, MAiNGO -- McCormick-based Algorithm for mixed-integer Nonlinear Global Optimization, Technical Report, Process Systems Engineering (AVT.SVT), RWTH Aachen University, 2018
• Pyomo, whose MC++ interface bounds factorable functions with McCormick relaxations (built on an earlier MC++ release)

License

MC++ is published under the Eclipse Public License. The bundled FADBAD++ (MC__USE_FADBAD, off by default) is a separate work distributed for non-commercial use only: commercial use requires a license from its authors.