Boscia.jl: A review and tutorial

TL;DR

This paper reviews the Boscia.jl framework for convex mixed-integer nonlinear optimization, highlighting its integration of Frank-Wolfe methods within a branch-and-bound approach and demonstrating its flexibility through practical examples.

Contribution

It provides a comprehensive overview and tutorial of the Boscia.jl framework, emphasizing its novel use of Frank-Wolfe methods and customizable features for solving convex MINLP problems.

Findings

Demonstrates the framework's flexibility through practical examples

Shows benefits of oracle-based access to objectives and gradients

Highlights the integration of Frank-Wolfe methods in MINLP solving

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.