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On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs.

Elisabeth Gaar1, Jon Lee2, Ivana Ljubić3

  • 1Institute of Production and Logistics Management, Johannes Kepler University Linz, Linz, Austria.

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Summary
This summary is machine-generated.

This study introduces disjunctive cuts (DCs) for integer bilevel programs with second-order cone constraints. The new methods enhance solution performance and outperform existing solvers for complex optimization problems.

Keywords:
Bilevel optimizationBranch-and-cutConic optimizationDisjunctive cutsNonlinear optimization

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Area of Science:

  • Optimization
  • Operations Research
  • Mathematical Programming

Background:

  • Integer bilevel programs with second-order cone constraints present significant computational challenges.
  • Existing methods struggle with the complexity of upper-level second-order cone constraints and lower-level convex-quadratic objectives.

Purpose of the Study:

  • To develop novel disjunctive cuts (DCs) for efficiently solving integer bilevel programs.
  • To propose effective DC separation strategies and methods for handling disjunctions and normalization.
  • To introduce a branch-and-cut algorithm and a cutting-plane method for specific problem variants.

Main Methods:

  • Development of a second-order-cone-based cut-generating procedure for disjunctive cuts.
  • Implementation of DC separation strategies, including redundancy removal and normalization.
  • Design of a branch-and-cut algorithm for general integer bilevel programs and a cutting-plane method for binary instances.

Main Results:

  • The proposed disjunctive cuts effectively separate bilevel-infeasible solutions.
  • Computational studies demonstrate significant performance improvements with the enhanced solution approaches.
  • The developed methods outperform a state-of-the-art generic solver on various problem instances.

Conclusions:

  • The novel disjunctive cut-based approaches provide a powerful tool for solving challenging integer bilevel programs.
  • The proposed algorithms offer superior performance compared to existing generic solvers.
  • The study highlights the effectiveness of disjunctive cuts in handling complex bilevel optimization problems.