Pseudo basic steps: Bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming

TL;DR

This paper introduces the pseudo basic step in convex disjunctive programming, providing guaranteed bounds on relaxation improvements using Lagrangian multipliers, with practical benefits demonstrated on clustering problems.

Contribution

It proposes a novel pseudo basic step operation with provable bounds, enhancing solution quality in convex disjunctive programming.

Findings

Significant bound improvements on K-means clustering instances.

Effective exploitation of bounds in solving convex MINLPs.

Practical benefits demonstrated through numerical examples.

Abstract

An elementary, but fundamental, operation in disjunctive programming is a basic step, which is the intersection of two disjunctions to form a new disjunction. Basic steps bring a disjunctive set in regular form closer to its disjunctive normal form and, in turn, produce relaxations that are at least as tight. An open question is: What are guaranteed bounds on the improvement from a basic step? In this paper, using properties of a convex disjunctive program's hull reformulation and multipliers from Lagrangian decomposition, we introduce an operation called a pseudo basic step and use it to provide provable bounds on this improvement along with techniques to exploit this information when solving a disjunctive program as a convex MINLP. Numerical examples illustrate the practical benefits of these bounds. In particular, on a set of K-means clustering instances, we make significant bound improvements relative to state-of-the-art commercial mixed-integer programming solvers.