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Effective anytime algorithm for multiobjective combinatorial optimization problems

Miguel Ángel Domínguez-Ríos, Francisco Chicano, Enrique Alba

TL;DR

A new exact anytime algorithm for multiobjective combinatorial optimization is proposed combining three novel ideas to enhance the anytime behavior using a set of 480 instances from different well-known benchmarks and four different performance measures.

Abstract

In multiobjective optimization, the result of an optimization algorithm is a set of efficient solutions from which the decision maker selects one. It is common that not all the efficient solutions can be computed in a short time and the search algorithm has to be stopped prematurely to analyze the solutions found so far. A set of efficient solutions that are well-spread in the objective space is preferred to provide the decision maker with a great variety of solutions. However, just a few exact algorithms in the literature exist with the ability to provide such a well-spread set of solutions at any moment: we call them anytime algorithms. We propose a new exact anytime algorithm for multiobjective combinatorial optimization combining three novel ideas to enhance the anytime behavior. We compare the proposed algorithm with those in the state-of-the-art for anytime multiobjective combinatorial optimization using a set of 480 instances from different well-known benchmarks and four different performance measures: the overall non-dominated vector generation ratio, the hypervolume, the general spread and the additive epsilon indicator. A comprehensive experimental study reveals that our proposal outperforms the previous algorithms in most of the instances.

Effective anytime algorithm for multiobjective combinatorial optimization problems

TL;DR

A new exact anytime algorithm for multiobjective combinatorial optimization is proposed combining three novel ideas to enhance the anytime behavior using a set of 480 instances from different well-known benchmarks and four different performance measures.

Abstract

In multiobjective optimization, the result of an optimization algorithm is a set of efficient solutions from which the decision maker selects one. It is common that not all the efficient solutions can be computed in a short time and the search algorithm has to be stopped prematurely to analyze the solutions found so far. A set of efficient solutions that are well-spread in the objective space is preferred to provide the decision maker with a great variety of solutions. However, just a few exact algorithms in the literature exist with the ability to provide such a well-spread set of solutions at any moment: we call them anytime algorithms. We propose a new exact anytime algorithm for multiobjective combinatorial optimization combining three novel ideas to enhance the anytime behavior. We compare the proposed algorithm with those in the state-of-the-art for anytime multiobjective combinatorial optimization using a set of 480 instances from different well-known benchmarks and four different performance measures: the overall non-dominated vector generation ratio, the hypervolume, the general spread and the additive epsilon indicator. A comprehensive experimental study reveals that our proposal outperforms the previous algorithms in most of the instances.
Paper Structure (26 sections, 6 theorems, 13 equations, 7 figures, 3 tables, 4 algorithms)

This paper contains 26 sections, 6 theorems, 13 equations, 7 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. First exact algorithms
  4. Algorithms based on recursion
  5. Modern algorithms
  6. Related algorithms that can be considered as anytime with good spread
  7. Background
  8. Definitions on multiobjective optimization
  9. Performance assessment
  10. General formulation for solving MOCO problems
  11. Mathematical program
  12. Selection of the next search zone to explore
  13. Updating the set of search zones
  14. Proposed anytime algorithm: TPA
  15. Alternation of directions in the search
  16. ...and 11 more sections

Key Result

Proposition 1

Given a box $B$, then $\left\{B_{i}(z) \right \}_{i=0}^{p}$ is a partition of $B$. When obtaining a non-dominated point $z$, the exploration of some other boxes in the future might produce the same solution. In order to prevent this, we split those boxes and remove the space weakly dominated by $z$, which is $B_0(z)$. That is why p-partition does not include $B_0(z)$. Figure fig:b It is very impo

Figures (7)

  • Figure 1: Box $\mathbf{B}=[(0,0,0),\,(20,15,10)]$ with non-dominated point $z=~(5,5,5)$ inside.
  • Figure 2: Resulting boxes after full p-split: $\mathbf{B_{1}}=~[(5,15,10)]$, $\mathbf{B_{2}}=~[(20,5,10)]$, and $\mathbf{B_{3}}=~[(20,15,5)]$. All the lower bounds are $(0,0,0)$. Each box shares some space with another box.
  • Figure 3: Resulting boxes after p-partition: $\mathbf{B^{*}_{1}}=[(0,5,5),\,(5,15,10)]$, $\mathbf{B^{*}_{2}}$$=[(0,0,5),\,(20,5,10)]$, and $\mathbf{B^{*}_{3}}=[(0,0,0),\,(20,15,5)]$. Each pair of boxes is disjoint.
  • Figure 5: Average rank values for the algorithms at 10 cut-points, using the metrics ONVGR and HVR. The lower the rank, the better the algorithm.
  • Figure 6: Average rank values for the algorithms at 10 cut-points, using the metrics $\Delta^{*}$ and $\varepsilon_{+}$. The lower the rank, the better the algorithm.
  • ...and 2 more figures

Theorems & Definitions (16)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Definition 3
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • ...and 6 more