# Benders Decomposition Using Graph Modeling and Multi-Parametric Programming

## TL;DR

This paper introduces a new framework for accelerating Benders decomposition using graph modeling and multiparametric programming surrogates to improve performance and interpretability in solving complex optimization problems.

## Contribution

The novel framework combines graph-theoretic modeling with multiparametric programming surrogates to replace subproblem solves with fast evaluations in Benders decomposition.

## Key findings

- Using mp surrogates achieves substantial speedups in subproblem solve time while preserving convergence guarantees.
- The framework enables solution analysis and interpretability through mp critical region tracking.
- The approach overcomes scalability issues of mp and supports heterogeneous subproblems with a unified structure.

## Abstract

Benders decomposition
is a widely used method for solving large
and structured optimization problems, but its performance is affected
by the repeated solution of subproblems. We propose a flexible and
modular algorithmic framework for accelerating Benders decomposition.
Specifically, we express the problem structure by using a graph-theoretic
modeling abstraction in which nodes represent optimization subproblems
and edges represent connectivity between subproblems. A key innovation
of our approach is that we embed multiparametric programming (mp)
surrogates for node subproblems, which maps the exact analytical map
of the subproblem solution space. The use of mp surrogates allows
us to replace subproblem solves with fast look-ups and function evaluations
for primal and dual variables during the iterative Benders process.
We formally show the equivalence between classical Benders cuts and
those derived from the mp solution. We implement our framework in
the open-source PlasmoBenders.jl software package.
To demonstrate the capabilities of the proposed framework, we apply
it to a two-stage stochastic programming problem, which aims to make
optimal capacity expansion decisions under market uncertainty. We
evaluate both single-cut and multicut variants of Benders decomposition
and show that the use of mp surrogates achieves substantial speedups
in subproblem solve time, while preserving the convergence guarantees
of Benders decomposition. We highlight advantages in solution analysis
and interpretability that is enabled by mp critical region tracking;
specifically, we show that these reveal how decisions evolve geometrically
across the Benders search. Our results aim to demonstrate that combining
surrogate modeling with graph modeling offers a promising and extensible
foundation for structure-exploiting decomposition. In addition, by
decomposing the problem into more tractable subproblems, the proposed
approach also aims to overcome scalability issues of mp. Finally,
the use of mp surrogates provides a unifying and modular optimization
framework that enables the representation of heterogeneous node subproblems
as modeling objects with a homogeneous structure.