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A High-Performance Parallel Algorithm for Multi-Objective Integer Optimization

Kathrin Prinz, Levin Nemesch, Stefan Ruzika

TL;DR

This work tackles the hardness of multi-objective integer optimization by introducing PEA, a parallel exact algorithm that exploits a tree-structured parameter space of lexicographic epsilon-constraint scalarizations to generate all nondominated images with largely autonomous, parallelizable tasks. The method leverages viable combinations and epsilon-components to cover the entire search space without excessive inter-thread communication, and it maintains a provable bound on the number of scalarization problems comparable to state-of-the-art sequential approaches, namely $O(|Y_N|^{\lfloor k/2 \rfloor})$. The contributions include extending PEA to any number of objectives, establishing a connection to local upper bounds, and proving that viable/true combinations form a directed tree that yields the full nondominated set when explored exhaustively. Computational results on knapsack and ILP instances show substantial parallel speedups, with PEA outperforming existing parallel methods and solving much larger instances within practical time limits, thereby offering a scalable tool for real-world MOIP problems.

Abstract

Multi-objective integer optimization problems are hard to solve, mainly because the number of nondominated images is often extremely large. We present the first exact algorithm, called PEA, that fully utilizes the multicore architecture of modern hardware. By exploiting the structure of the parameter set of the underlying scalarization, PEA can use a high number of threads while avoiding the usual pitfalls of parallel computing. It is highly scalable and easy to implement. As a result, PEA can solve much larger instances than previous state-of-the-art algorithms. Besides, PEA has a sound theoretical foundation. Unlike other existing parallel algorithms, it always solves the same number of scalarization problems as comparable sequential algorithms. We demonstrate the potential of PEA in a computational study.

A High-Performance Parallel Algorithm for Multi-Objective Integer Optimization

TL;DR

This work tackles the hardness of multi-objective integer optimization by introducing PEA, a parallel exact algorithm that exploits a tree-structured parameter space of lexicographic epsilon-constraint scalarizations to generate all nondominated images with largely autonomous, parallelizable tasks. The method leverages viable combinations and epsilon-components to cover the entire search space without excessive inter-thread communication, and it maintains a provable bound on the number of scalarization problems comparable to state-of-the-art sequential approaches, namely . The contributions include extending PEA to any number of objectives, establishing a connection to local upper bounds, and proving that viable/true combinations form a directed tree that yields the full nondominated set when explored exhaustively. Computational results on knapsack and ILP instances show substantial parallel speedups, with PEA outperforming existing parallel methods and solving much larger instances within practical time limits, thereby offering a scalable tool for real-world MOIP problems.

Abstract

Multi-objective integer optimization problems are hard to solve, mainly because the number of nondominated images is often extremely large. We present the first exact algorithm, called PEA, that fully utilizes the multicore architecture of modern hardware. By exploiting the structure of the parameter set of the underlying scalarization, PEA can use a high number of threads while avoiding the usual pitfalls of parallel computing. It is highly scalable and easy to implement. As a result, PEA can solve much larger instances than previous state-of-the-art algorithms. Besides, PEA has a sound theoretical foundation. Unlike other existing parallel algorithms, it always solves the same number of scalarization problems as comparable sequential algorithms. We demonstrate the potential of PEA in a computational study.
Paper Structure (15 sections, 18 theorems, 54 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 15 sections, 18 theorems, 54 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature
  3. Our Contribution
  4. Preliminaries
  5. The Lexicographic Epsilon-Constraint Scalarization
  6. Epsilon-Components
  7. Viable Combinations
  8. Viable Combinations under General Position
  9. Viable Combinations and Upper Bounds
  10. Viable Combinations in the Generic Case
  11. Algorithm
  12. Improvements
  13. Computational Study
  14. Results
  15. Conclusions

Key Result

Lemma 3

Let $y\in Y_N$ and $\varepsilon\in E(y)$. Then, the following holds:

Figures (8)

  • Figure 1: A visualization of \ref{['ex:introductory4D']}: The projected nondominated images in $\mathbb{R}^3$ and their epsilon-components.
  • Figure 2: The projected nondominated images and dummy images as well as their epsilon-components of \ref{['ex:introductory4D']} in $\mathbb{R}^3$. All viable parameters are marked. The four viable parameters that describe the epsilon-component of a nondominated image are marked bigger and in their respective color. The other seven parameters describe the epsilon-component of the dummy image $d^4$, i. e., the part of the parameter space for which the respective epsilon-constraint scalarization problems are infeasible.
  • Figure 3: The tree as induced by the order described in \ref{['def:scions']}, i. e., $G=(V,A)$ where $V=\mathcal{V}(Y_N)$ and $A=\{(\mathcal{Y},\mathcal{Z}): \mathcal{Z}=\mathrm{scion}^{\ell}(\mathcal{Y}) \text{ for some } \ell\in[k-1]\}$, for the nondominated set given in \ref{['ex:introductory4D']}.
  • Figure 4: We consider a tri-objective problem for which the nondominated set is given by $Y_N = \left\{ y^1= (5,4,1)^\top, y^2=(2,6,2)^\top, y^3=(6,2,4)^\top, y^4=(3,3,5)^\top\right\}.$ Then, the construction of $\mathcal{Y}(y^4)$ as in the proof of \ref{['thm: viable combination construction']} can be interpreted as follows: Starting with $(y^4_1+\delta,y^4_2+\delta)$, we shoot a ray in direction $(1,0)$ until we "hit" the first epsilon-component, $E(y^3)$. Then, we "shoot" another ray in direction $(0,1)$ until we hit another epsilon component, $E(y^1)$. Thus, $\mathcal{Y}(y^4)=(y^3,y^1)$.
  • Figure 5: The left images depicts the epsilon-components of the nondominated set $Y_N$ of \ref{['ex:true combinations']}. All viable parameters are marked. The right image depicts the same but for the set $\Phi(Y_N)$. The parameter that is marked in green in the left image is defined by three different viable combinations, $(y^1,y^3),(y^2,y^1)$ and $(y^2,y^3)$. Only two of these are also viable combinations of $\Phi(Y_N)$, $(\Phi(y^1),\Phi(y^3))$ and $(\Phi(y^2), \Phi(y^1))$. $(\Phi(y^1),\Phi(y^3))$ and $(\Phi(y^2), \Phi(y^1))$ define different viable parameters, both are marked in green in the right image.
  • ...and 3 more figures

Theorems & Definitions (47)

  • Definition 1
  • Example 2
  • Lemma 3
  • proof
  • Definition 4
  • Definition 5
  • Example 6
  • Remark 7
  • Corollary 8
  • proof
  • ...and 37 more