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A Unifying Complexity-Certification Framework for Branch-and-Bound Algorithms for Mixed-Integer Linear and Quadratic Programming

Shamisa Shoja, Daniel Arnström, Daniel Axehill

TL;DR

This work develops a unifying complexity-certification framework for branch-and-bound solvers tackling multi-parametric MILPs and MIQPs, enabling exact worst-case bounds on iterations and relaxations across parameter regions. It integrates branching, node-selection, and primal heuristics, and introduces conservative quadratic-function handling (via affine approximations, McCormick relaxations, and regions-as-atoms) to maintain tractable, polyhedral partitions in MIQPs. The framework proves equivalence with online B&B under precise assumptions, and extends to warm-starting and suboptimal variants, providing reliable upper bounds on online computational effort relevant to real-time hybrid MPC. Numerical experiments on random mp-MILPs/MIQPs and an MPC example demonstrate the framework’s ability to yield tight, conservative complexity guarantees and actionable insights for solver customization. Overall, the methodology enhances reliability and predictability of B&B-based solvers for real-time, parameter-varying optimization in hybrid systems.

Abstract

In model predictive control (MPC) for hybrid systems, solving optimization problems efficiently and with guarantees on worst-case computational complexity is critical to satisfy the real-time constraints in these applications. These optimization problems often take the form of mixed-integer linear programs (MILPs) or mixed-integer quadratic programs (MIQPs) that depend on system parameters. A common approach for solving such problems is the branch-and-bound (B&B) method. This paper extends existing complexity certification methods by presenting a unified complexity-certification framework for B&B-based MILP and MIQP solvers, specifically for the family of multi-parametric MILP and MIQP problems that arise in, e.g., hybrid MPC applications. The framework provides guarantees on worst-case computational measures, including the maximum number of iterations or relaxations B&B algorithms require to reach optimality. It systematically accounts for different branching and node selection strategies, as well as heuristics integrated into B&B, ensuring a comprehensive certification framework. By offering theoretical guarantees and practical insights for solver customization, the proposed framework enhances the reliability of B&B for real-time application. The usefulness of the proposed framework is demonstrated through numerical experiments on both random MILPs and MIQPs, as well as on MIQPs arising from a hybrid MPC problem.

A Unifying Complexity-Certification Framework for Branch-and-Bound Algorithms for Mixed-Integer Linear and Quadratic Programming

TL;DR

This work develops a unifying complexity-certification framework for branch-and-bound solvers tackling multi-parametric MILPs and MIQPs, enabling exact worst-case bounds on iterations and relaxations across parameter regions. It integrates branching, node-selection, and primal heuristics, and introduces conservative quadratic-function handling (via affine approximations, McCormick relaxations, and regions-as-atoms) to maintain tractable, polyhedral partitions in MIQPs. The framework proves equivalence with online B&B under precise assumptions, and extends to warm-starting and suboptimal variants, providing reliable upper bounds on online computational effort relevant to real-time hybrid MPC. Numerical experiments on random mp-MILPs/MIQPs and an MPC example demonstrate the framework’s ability to yield tight, conservative complexity guarantees and actionable insights for solver customization. Overall, the methodology enhances reliability and predictability of B&B-based solvers for real-time, parameter-varying optimization in hybrid systems.

Abstract

In model predictive control (MPC) for hybrid systems, solving optimization problems efficiently and with guarantees on worst-case computational complexity is critical to satisfy the real-time constraints in these applications. These optimization problems often take the form of mixed-integer linear programs (MILPs) or mixed-integer quadratic programs (MIQPs) that depend on system parameters. A common approach for solving such problems is the branch-and-bound (B&B) method. This paper extends existing complexity certification methods by presenting a unified complexity-certification framework for B&B-based MILP and MIQP solvers, specifically for the family of multi-parametric MILP and MIQP problems that arise in, e.g., hybrid MPC applications. The framework provides guarantees on worst-case computational measures, including the maximum number of iterations or relaxations B&B algorithms require to reach optimality. It systematically accounts for different branching and node selection strategies, as well as heuristics integrated into B&B, ensuring a comprehensive certification framework. By offering theoretical guarantees and practical insights for solver customization, the proposed framework enhances the reliability of B&B for real-time application. The usefulness of the proposed framework is demonstrated through numerical experiments on both random MILPs and MIQPs, as well as on MIQPs arising from a hybrid MPC problem.

Paper Structure

This paper contains 46 sections, 16 theorems, 18 equations, 3 figures, 2 tables, 14 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Branch and bound
  4. Introduction
  5. Algorithm overview
  6. Cut-condition evaluation
  7. Branching strategies
  8. Node-selection strategies
  9. Heuristics in B&B
  10. Start heuristics
  11. Improvement heuristics
  12. Integration of heuristics into the online B&B algorithm
  13. Unified complexity-certification framework
  14. Algorithm overview
  15. Cut-condition evaluation
  16. ...and 31 more sections

Key Result

Lemma 1

Let $\stackunder[1.2pt]{$J$}{}_{\eta}$ denote the objective function value of a relaxation at node $\eta$, and let $\hat{\eta} \in \mathcal{D}(\eta)$. Then $\stackunder[1.2pt]{$J$}{}_{\hat{\eta}} \geq \stackunder[1.2pt]{$J$}{}_{\eta}$.

Figures (3)

  • Figure 1: Decomposition of a polyhedral region $\Theta$ after relaxation certification using solveCert in Algorithm \ref{['alg:cert_uni']}. Each subregion is potentially further decomposed by cutCert.
  • Figure 2: Resulting parameter space for a random example determined by (a) applying Algorithm \ref{['alg:cert_uni']}; (b) executing Algorithm \ref{['alg:BnB_on_uni']} over a deterministic grid in the parameter space.
  • Figure 3: Regulating the inverted pendulum on a cart with contact forces.

Theorems & Definitions (31)

  • Definition 1: Node
  • Lemma 1
  • proof
  • Definition 2
  • Definition 3
  • Remark 1
  • Remark 2
  • Lemma 2: Equivalence of sortCert and sort
  • Lemma 3: Equivalence of mostInfScoreCert and \ref{['eq:branch_minf_on_uni']}
  • Lemma 4
  • ...and 21 more