Exact Decomposition Branching exploiting Lattice Structures

TL;DR

This paper introduces a novel rounding rule based on lattice structures to ensure exact solutions in mixed-integer linear optimization problems with strict inequalities, improving convergence and reducing computational complexity.

Contribution

It proposes a new lattice-based rounding rule for strict inequalities and integrates it into Decomposition Branching, guaranteeing finite termination with exact solutions.

Findings

Enhanced algorithm terminates finitely with exact solutions

Typically produces smaller branch-and-bound trees

Validated through computational experiments on models with detectable ta-regularity

Abstract

Strict inequalities in mixed-integer linear optimization can cause difficulties in guaranteeing convergence and exactness. Utilizing that optimal vertex solutions follow a lattice structure we propose a rounding rule for strict inequalities that guaranties exactness. The lattice used is generated by $\Delta$-regularity of the constraint matrix belonging to the continuous variables. We apply this rounding rule to Decomposition Branching by Yildiz et al., which uses strict inequalities in its branching rule. We prove that the enhanced algorithm terminates after finite many steps with an exact solution. To validate our approach, we conduct computational experiments for two different models for which $\Delta$-regularity is easily detectable. The results confirm the exactness of our enhanced algorithm and demonstrate that it typically generates smaller branch-and-bound trees.