MDOToolbox: Multidisciplinary Design Optimization with Bayesian Collaborative Frameworks

MDOToolbox is a Python library for solving multi-disciplinary optimization (MDO) problems using hierarchical decomposition and Bayesian optimization. It provides five optimization frameworks, Gaussian Process surrogates, and a suite of built-in benchmark problems for rapid prototyping and research.

Cite us (preprint)

@misc{baco2026,
      title         = {Bayesian Algorithm for Collaborative Optimization with Application to Aircraft Design}, 
      author        = {Mohamed Ali Belhafnaoui and Youssef Diouane},
      year          = {2026},
      eprint        = {2605.05474},
      archivePrefix = {arXiv},
      primaryClass  = {math.OC},
      url           = {https://arxiv.org/abs/2605.05474}, 
}

Paper PDF

Features

• Multiple MDO frameworks CO, BACO, ICO, MCO, and ECO for different problem classes
• Bayesian optimization with Gaussian Process surrogates for expensive black-box functions
• 7 built-in benchmark problems covering analytical, mechanical, thermal, biomedical, and chemical domains
• Parallel multi-start acquisition optimization for robust global search
• Support for constrained and unconstrained optimization

Installation

Basic install:

pip install mdotoolbox

Development install (with test dependencies):

pip install -e ".[test]"

Dependencies: numpy, scipy, smt, joblib, numba, matplotlib, scikit-learn, gpytorch, torch, jaxtyping, typeguard, pandas, openmdao, openaerostruct, pymc.

Quick Start

This example solves the classic Sellar benchmark problem using the Bayesian Collaborative Optimization (BACO) framework:

import numpy as np
from mdotoolbox.benchmarking import sellar_problem
from mdotoolbox.frameworks import (
    BayesianCollaborativeOptimization,
    BACOSubsystem,
    BACOSystem,
)

# Load benchmark problem
problem_dict = sellar_problem()

# Build system
subsystems = [BACOSubsystem(**sub) for sub in problem_dict["subsystems"]]
system = BACOSystem(
    problem=problem_dict["system_problem"],
    subsystems=subsystems,
)

# Create and run solver
solver = BayesianCollaborativeOptimization(
    system=system,
    subsystem_optimizer="cobyqa",
    system_optimizer="cobyqa",
    epsilon_J=1e-6,
    epsilon_h=1e-6,
    max_eval=50,
    n_multistart=5,
    n_initial=8,
)

z0 = np.array([5.0, 2.0])
x0 = np.array([1.0])
y0 = np.array([1.0, 1.0])
results = solver.solve(z0, x0, y0)
print(results)

Framework Selection Guide

Framework	Description	Best For
CO (Collaborative Optimization)	Classical bilevel formulation	Well-behaved problems; fastest when it works
BACO (Bayesian Collaborative Optimization)	CO with GP surrogates	Expensive simulations; efficient with function evaluations
ICO (Improved Collaborative Optimization)	Enhanced convergence	When CO has convergence issues
MCO (Modified Collaborative Optimization)	Alternative formulation	Experimental problems
ECO (Enhanced Collaborative Optimization)	Advanced convergence criteria	Best convergence guarantees; slowest but most reliable

Built-in Benchmark Problems

Function	Description
sellar_problem()	Classic 2-discipline analytical benchmark
scalable_mdo_problem()	Configurable scalable problem with adjustable size
speed_reducer_problem()	Mechanical engineering design problem
electronic_packaging_problem()	Thermal management problem
heart_dipole_problem()	Biomedical application
power_converter_problem()	Power electronics design
propane_combustion_problem()	Chemical process optimization

All benchmark functions return a dictionary with system_problem and subsystems keys, ready to pass into any framework.

Project Structure

.
├── src/mdotoolbox/   # Core library source
├── docs/             # Documentation source
├── pyproject.toml    # Build and dependency configuration
└── README.md         # Project overview

Contributing

See CONTRIBUTING.md for development setup, testing instructions, and contribution guidelines.

License

Apache License 2.0 — see LICENSE for details.

Links

• Documentation: https://moebehfn.github.io/mdotoolbox/
• Repository: https://github.com/moebehfn/mdotoolbox
• Author: Mohamed Ali Belhafnaoui (mohamed-ali.belhafnaoui@etud.polymtl.ca)

Please consider supporting our other project: fluxwing

Python library for simulating the aerodynamics of a wing at a given condition using VLM, as well as the structural loads using an Euler-Bernoulli solver:

• github
• documentation