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

Dimitri J. Papageorgiou, Francisco Trespalacios

TL;DR

This paper develops provable bound improvements for convex disjunctive programs by introducing pseudo basic steps, guided by a partial Lagrangian relaxation and partition relaxations. It formalizes how aggregated Lagrange multipliers yield a nonnegative improvement over the base hull relaxation and compares pseudo steps to actual steps through illustrative and computational evidence. The authors validate the approach on conic-quadratic disjunctive programs and, notably, on K-means clustering formulated as MIQCP, where partition relaxations (TresPapas) substantially tighten bounds beyond commercial solvers. The work suggests promising directions for step selection strategies, integration of cuts, and extending the methodology to broader convex MINLPs, potentially enabling more scalable exact solutions in practice.

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.

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

TL;DR

This paper develops provable bound improvements for convex disjunctive programs by introducing pseudo basic steps, guided by a partial Lagrangian relaxation and partition relaxations. It formalizes how aggregated Lagrange multipliers yield a nonnegative improvement over the base hull relaxation and compares pseudo steps to actual steps through illustrative and computational evidence. The authors validate the approach on conic-quadratic disjunctive programs and, notably, on K-means clustering formulated as MIQCP, where partition relaxations (TresPapas) substantially tighten bounds beyond commercial solvers. The work suggests promising directions for step selection strategies, integration of cuts, and extending the methodology to broader convex MINLPs, potentially enabling more scalable exact solutions in practice.

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.
Paper Structure (13 sections, 7 theorems, 34 equations, 4 figures, 6 tables, 2 algorithms)

This paper contains 13 sections, 7 theorems, 34 equations, 4 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Basic Steps
  3. Main results
  4. Partition relaxation and guaranteed bound improvements from basic steps
  5. Illustrative example
  6. Discussion
  7. Computational experiments
  8. Conic quadratic disjunctive programs
  9. $K$-means clustering
  10. MIQCP formulations
  11. Partition relaxation
  12. $K$-means computational experiments
  13. Conclusions and future research directions

Key Result

Proposition 1

For $k=1,2$, let $\mathcal{F}_k = \cup_{i \in \mathcal{D}_k} \mathcal{C}_{ki}$, where each $\mathcal{C}_{ki}$ is a compact convex set. Then if and only if every extreme point of $\mathcal{T}_R$ is an extreme point of $\mathcal{C}_{1p} \cap \mathcal{C}_{2q}$ for some $(p,q) \in \mathcal{D}_1 \times \mathcal{D}_2$.

Figures (4)

  • Figure 1: Illustration of basic steps.
  • Figure 2: Illustration of example \ref{['model:simplest_example']}. a) Shows the feasible region and optimal solution. b) Shows the continuous relaxation of the hull reformulation, projected onto the original space.
  • Figure 3: Illustration of the hull relaxation after basic steps for (\ref{['model:simplest_example']}) intersecting a) disjunctions 1 and 2; b) disjunctions 1 and 3; and c) disjunctions 2 and 3
  • Figure 4: Comparison of average optimality gap closed for various solvers and methods on $K$-means instances.

Theorems & Definitions (12)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Corollary 1
  • Corollary 2
  • Remark 2
  • Remark 3
  • Remark 4
  • ...and 2 more