Boscia.jl: A review and tutorial

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, showcases extensions post-publication 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. Additionally, ablation studies are performed on the three examples to investigate the performance impact of the different features and customizations. The aim of this tutorial is to provide readers with an understanding of the main principles of the framework.

38 pages, 11 figures. 3 tables