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Surrogate "Level-Based" Lagrangian Relaxation for mixed-integer linear programming.

Mikhail A Bragin1, Emily L Tucker2

  • 1Department of Electrical and Computer Engineering, University of Connecticut, 371 Fairfield Way, U-4157, Storrs, 06269, CT, USA. mikhail.bragin@uconn.edu.

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A novel price-based decomposition method efficiently solves complex Mixed-Integer Linear Programming (MILP) problems. This approach uses a unique decision-based stepsizing strategy, achieving significant speedups and optimal solutions for large-scale optimization challenges.

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Area of Science:

  • Operations Research
  • Computer Science
  • Applied Mathematics

Background:

  • Mixed-Integer Linear Programming (MILP) is crucial in science and industry but faces combinatorial complexity.
  • Increasing problem size leads to super-linear growth in solutions and computational effort.

Purpose of the Study:

  • To develop an efficient computational method for solving complex MILP problems.
  • To exploit decomposition for complexity reduction and Polyak's stepsizing for fast coordination.

Main Methods:

  • A price-based decomposition and coordination approach is introduced.
  • Novel, decision-based stepsizes are inferred by solving an auxiliary constraint satisfaction problem.
  • The method leverages super-linear complexity reduction and geometric convergence properties.

Main Results:

  • Certifiably optimal solutions were obtained for most large-scale Generalized Assignment Problems.
  • A two-order-of-magnitude speedup was observed compared to Branch-and-Cut for scheduling problems.
  • The method demonstrates significant computational efficiency for complex MILP instances.

Conclusions:

  • The developed price-based method offers a computationally efficient way to resolve complex MILP problems.
  • This approach has a major impact on various scientific fields requiring MILP solutions.
  • It provides a novel, decision-based alternative to heuristic stepsize setting.