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Exact Decomposition Branching exploiting Lattice Structures

Katrin Halbig, Timm Oertel, Dieter Weninger

Abstract

Strict inequalities in mixed-integer linear optimization can cause difficulties in guaranteeing convergence and exactness. Utilizing that optimal vertex solutions follow a lattice structure we propose a rounding rule for strict inequalities that guaranties exactness. The lattice used is generated by $Δ$-regularity of the constraint matrix belonging to the continuous variables. We apply this rounding rule to Decomposition Branching by Yildiz et al., which uses strict inequalities in its branching rule. We prove that the enhanced algorithm terminates after finite many steps with an exact solution. To validate our approach, we conduct computational experiments for two different models for which $Δ$-regularity is easily detectable. The results confirm the exactness of our enhanced algorithm and demonstrate that it typically generates smaller branch-and-bound trees.

Exact Decomposition Branching exploiting Lattice Structures

Abstract

Strict inequalities in mixed-integer linear optimization can cause difficulties in guaranteeing convergence and exactness. Utilizing that optimal vertex solutions follow a lattice structure we propose a rounding rule for strict inequalities that guaranties exactness. The lattice used is generated by -regularity of the constraint matrix belonging to the continuous variables. We apply this rounding rule to Decomposition Branching by Yildiz et al., which uses strict inequalities in its branching rule. We prove that the enhanced algorithm terminates after finite many steps with an exact solution. To validate our approach, we conduct computational experiments for two different models for which -regularity is easily detectable. The results confirm the exactness of our enhanced algorithm and demonstrate that it typically generates smaller branch-and-bound trees.

Paper Structure

This paper contains 12 sections, 9 theorems, 29 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Rounding Approach based on Lattices
  3. Determination of $\Delta$-Regularity
  4. Enhancement of Decomposition Branching
  5. Properties of the initial Algorithm
  6. Exactness through Exploitation of $\Delta$-Regularity
  7. Proof of Exactness and Termination
  8. Computational Study
  9. Computational Setup
  10. Test Sets
  11. Performance results
  12. Conclusion

Key Result

Lemma 3.1

Let $A$ be an integral matrix. Its minimal $\Delta$-regularity is bounded from below by the least common multiple (lcm) of all entries of $A$.

Figures (4)

  • Figure 1: Rounding of strict inequalities based on $\Delta$-regularity.
  • Figure 2: Example of initial Decomposition Branching. For each block, the corresponding polyhedron is shown as a grey shaded area with the LP solution of the first iteration in blue and the optimal solution in red.
  • Figure 3: Number of nodes used by $\Delta$DB compared to DB with four different $\varepsilon$ for test set MISL. Shapes define the state of the two compared variants.
  • Figure 4: Number of nodes used by $\Delta$DB compared to DB with four different $\varepsilon$ for test set CFL. Shapes define the state of the two compared variants.

Theorems & Definitions (18)

  • Definition 2.1: Minimal $\Delta$-Regularity
  • Lemma 3.1: Lower Bound Minimal $\Delta$-Regularity
  • Lemma 3.2: Upper Bound Minimal $\Delta$-Regularity
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 3.6
  • ...and 8 more