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Boscia.jl: A review and tutorial

Wenjie Xiao, Deborah Hendrych, Mathieu Besançon, Sebastian Pokutta

TL;DR

Boscia.jl presents a framework for convex MINLPs where the nonlinearity lies in the objective, combining a branch-and-bound backbone with Frank-Wolfe subproblem solvers that operate over $\mathrm{conv}(\mathcal{X})$ and rely on an LMO that can propagate integrality. The framework supports a diverse set of FW variants, configurable node strategies, and multiple LMO implementations (including fully custom, structure-aware LMOs), enabling inexact node processing, warm-starting, and exploitation of combinatorial structure. Through three tutorials—network design, graph isomorphism, and optimal experiment design—the paper demonstrates practical workflows, custom LMO development, and domain-aware adaptations that yield efficient, scalable solutions for large, structured MINLPs. The work highlights Boscia's potential to outperform traditional outer-approximation methods in problems with exploitable combinatorial constraints, while outlining future directions such as stochastic objectives, non-convex objectives with spatial branching, and broader LMO support. These contributions offer a flexible, pragmatic toolkit for practitioners needing tailored MINLP solving in domains with rich combinatorial structure.

Abstract

Mixed-integer nonlinear optimization (MINLP) comprises a large class of problems that are challenging to solve and exhibit a wide range of structures. The Boscia framework Hendrych et al. (2025b) focuses on convex MINLP where the nonlinearity appears in the objective only. This paper provides an overview of the framework and practical examples to illustrate its use and customizability. One key aspect is the integration and exploitation of Frank-Wolfe methods as continuous solvers within a branch-and-bound framework, enabling inexact node processing, warm-starting and explicit use of combinatorial structure among others. Three examples illustrate its flexibility, the user control over the optimization process and the benefit of oracle-based access to the objective and its gradient. The aim of this tutorial is to provide readers with an understanding of the main principles of the framework.

Boscia.jl: A review and tutorial

TL;DR

Boscia.jl presents a framework for convex MINLPs where the nonlinearity lies in the objective, combining a branch-and-bound backbone with Frank-Wolfe subproblem solvers that operate over and rely on an LMO that can propagate integrality. The framework supports a diverse set of FW variants, configurable node strategies, and multiple LMO implementations (including fully custom, structure-aware LMOs), enabling inexact node processing, warm-starting, and exploitation of combinatorial structure. Through three tutorials—network design, graph isomorphism, and optimal experiment design—the paper demonstrates practical workflows, custom LMO development, and domain-aware adaptations that yield efficient, scalable solutions for large, structured MINLPs. The work highlights Boscia's potential to outperform traditional outer-approximation methods in problems with exploitable combinatorial constraints, while outlining future directions such as stochastic objectives, non-convex objectives with spatial branching, and broader LMO support. These contributions offer a flexible, pragmatic toolkit for practitioners needing tailored MINLP solving in domains with rich combinatorial structure.

Abstract

Mixed-integer nonlinear optimization (MINLP) comprises a large class of problems that are challenging to solve and exhibit a wide range of structures. The Boscia framework Hendrych et al. (2025b) focuses on convex MINLP where the nonlinearity appears in the objective only. This paper provides an overview of the framework and practical examples to illustrate its use and customizability. One key aspect is the integration and exploitation of Frank-Wolfe methods as continuous solvers within a branch-and-bound framework, enabling inexact node processing, warm-starting and explicit use of combinatorial structure among others. Three examples illustrate its flexibility, the user control over the optimization process and the benefit of oracle-based access to the objective and its gradient. The aim of this tutorial is to provide readers with an understanding of the main principles of the framework.

Paper Structure

This paper contains 10 sections, 17 equations, 4 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. The Boscia framework
  3. Frank-Wolfe variants
  4. The LMO with integer constraints
  5. Optional settings
  6. Examples and tutorials
  7. Network design problem - simple LMO and MOI LMO
  8. Graph isomorphism problem - creating a self-managed LMO
  9. Optimal design of experiments - restricted function domains
  10. Discussion and conclusion

Figures (4)

  • Figure 1: The schematic of the algorithm including the optimization problems solved at the different layers.
  • Figure 2: A traffic flow optimization example reproduced from 2024_SharmaHendrychBesanconPokutta_NetworkdesignMicoFrankwolfe showing initial and optimized flow distributions. The line thickness represents flow magnitude, with dashed lines indicating optional edges that can be added.
  • Figure 3: Three isomorphic representations of the Petersen graph adapted from the example in hasan2017graphettes. All three layouts depict the same abstract graph through different planar embeddings. They preserve identical adjacency structure and graph-theoretic invariants, illustrating the concept of graph isomorphism.
  • Figure 4: Progress of the upper and lower bounds of the B&B tree in Boscia for two instances of OEDP.