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Evolutionary Multitasking for Multiobjective Continuous Optimization: Benchmark Problems, Performance Metrics and Baseline Results

Yuan Yuan, Yew-Soon Ong, Liang Feng, A. K. Qin, Abhishek Gupta, Bingshui Da, Qingfu Zhang, Kay Chen Tan, Yaochu Jin, Hisao Ishibuchi

arXiv:1706.02766v1cs.NE

TL;DR

MO-MFO solves multiple multiobjective tasks simultaneously, motivating evaluation across differing task relationships. This report defines nine two-task benchmarks and finds that MO-MFEA generally outperforms NSGA-II, with NILS the overall-performance exception.

  • Problem

    Understanding algorithm behavior where evolutionary multitasking and multiobjective optimization meet motivates evaluating MO-MFO across differing task relationships.

    Source: §I · p. 1

  • Method

    The report defines nine two-task benchmarks spanning global-minima intersection and fitness-landscape similarity, and proposes a comprehensive IGD-based ranking criterion.

    Source: §II · p. 2, B. Ranking of the MO-MFO Algorithms · p. 5

  • Results

    MO-MFEA performs significantly better than NSGA-II on both tasks in most test problems, and its overall performance is better except on NILS.

    Source: A. Results of IGD and MSS · p. 7

  • Takeaways & Limitations

    The benchmarks and ranking criterion support comprehensive evaluation of MO-MFO algorithms across task relationships.

    Source: §II · p. 2, B. Ranking of the MO-MFO Algorithms · p. 5

    The synthetic benchmarks omit phenotypic overlap because their functions lack substantial contextual meaning for that condition.

    Source: §II · p. 2

Abstract

from arXiv · show

In this report, we suggest nine test problems for multi-task multi-objective optimization (MTMOO), each of which consists of two multiobjective optimization tasks that need to be solved simultaneously. The relationship between tasks varies between different test problems, which would be helpful to have a comprehensive evaluation of the MO-MFO algorithms. It is expected that the proposed test problems will germinate progress the field of the MTMOO research.

I. INTRODUCTION

The introduction frames MO-MFO as solving multiple multiobjective tasks simultaneously with one evolving population and proposes nine benchmark problems. It summarizes their properties, including task similarity, and presents four kinds of Pareto fronts.

I. INTRODUCTION

  • MO-MFO combines evolutionary multitasking and multiobjective optimization, solving multiple multiobjective tasks simultaneously with one evolving EA population; it was first discussed in.The proof of concept in used two synthetic benchmarks and a real-world case study.Source: §I · p. 1

II. DEFINITIONS OF BENCHMARK PROBLEMS

  • Task relationships are characterized through q(x) functions, with landscape similarity measured by Spearman rank correlation over 1,000,000 randomly sampled points.The report treats a multiobjective task’s Pareto-optimal solutions as attained iff its corresponding q(x) reaches its global minimum; similarity bands are high for (0, 1/3], medium for (1/3, 2/3], and low for (2/3, 1].Source: §II · p. 2
  • Nine benchmarks cross complete, partial, and no global-minimum intersection with high, medium, and low landscape similarity; phenotypic overlap is excluded for synthetic tasks.The report omits phenotypic overlap because synthetic benchmark functions lack substantial contextual meaning for decision variables.Source: §II · p. 2
  • Table I summarizes the test problems and their properties, including task similarity sim(T1, T2), while Fig. 1 shows four kinds of Pareto fronts.The passage defines sim(T1, T2) as the similarity between two tasks in a test problem.Source: §II · p. 5, Figure 1 · p. 5

III. PERFORMANCE EVALUATION

Section III describes the evaluation process and treats the test problems as black-box problems whose analytic forms are unknown to the algorithms.

III. PERFORMANCE EVALUATION

  • The evaluation treats each test problem as a black box, so its analytic form is unavailable to the algorithm.The section describes this as part of the process for evaluating algorithm performance on the proposed problems.Source: §III · p. 5

A. Performance Metric

IGD evaluates an algorithm’s performance on each task using its approximate nondominated set and a uniformly distributed reference set over that task’s Pareto front.

A. Performance Metric

  • IGD compares the task’s nondominated objective vectors with a uniformly distributed reference set over its Pareto front, after normalization using the reference set’s objective extrema.The approximate set is A, and the reference set is P*; the reference set is normalized alongside A.Source: A. Performance Metric · p. 5
  • A small IGD indicates that the approximate set is close to the reference front and does not miss its parts, reflecting convergence and diversity when P* sufficiently represents the front.The distance used is Euclidean distance in normalized objective space.Source: A. Performance Metric · p. 5
  • The reference-set data and IGD computation source code are available in the provided zip package.Source: A. Performance Metric · p. 5

B. Ranking of the MO-MFO Algorithms

The report proposes the mean standard score (MSS) to combine task-level IGD results into an overall ranking of MO-MFO algorithms on each test problem.

B. Ranking of the MO-MFO Algorithms

  • MSS provides a comprehensive criterion for ranking overall MO-MFO performance on a test problem, with smaller values indicating better performance.Source: B. Ranking of the MO-MFO Algorithms · p. 5
  • MSS is computed from each task’s IGD value standardized using that task’s mean and standard deviation.The task-specific means and standard deviations are calculated from the IGD results of all algorithms across all runs.Source: B. Ranking of the MO-MFO Algorithms · p. 5

1) The maximum number of nondominated objective vectors in the approximate set:

The evaluation caps the approximate set used to compute IGD at different sizes depending on the task’s objective count.

1) The maximum number of nondominated objective vectors in the approximate set:

2) The maximum number of function evaluations:

The maximum is 200,000 function evaluations for every test problem, with evaluations counted per task and not distinguished across tasks.

2) The maximum number of function evaluations:

3) The number of independent runs:

Each algorithm is run independently 30 times on every multiobjective multitasking test problem.

3) The number of independent runs:

4) The parameter settings for the algorithm:

Parameter settings are shared within two groups: NIMS/NILS and the remaining test problems.

4) The parameter settings for the algorithm:

IV. BASELINE RESULTS

The baseline section compares MO-MFEA with NSGA-II, which solves the two tasks separately rather than simultaneously in one population.

IV. BASELINE RESULTS

  • MO-MFEA is compared with NSGA-II, which solves the two tasks separately while MO-MFEA solves them simultaneously in one population.Source: §IV · pp. 6–7
  • NSGA-II uses a maximum of 100,000 evaluations per task, whereas MO-MFEA uses 200,000 per two-task test problem.The budgets are set to make the comparison fair.Source: §IV · p. 7
  • Both algorithms use simulated binary crossover and polynomial mutation; Table II lists their crossover and mutation parameters.Source: §IV · p. 7, Table II · p. 7

A. Results of IGD and MSS

MO-MFEA generally outperforms NSGA-II on the proposed test problems, with exceptions on some per-task IGD comparisons and on NILS overall. It also converges faster on most problems and usually yields better Pareto-front approximations.

A. Results of IGD and MSS

  • MO-MFEA performs significantly better on both tasks in all problems except PILS and NILS; on PILS it is slightly worse on T1 but much better on T2.MO-MFEA and NSGA-II have very close performance on both NILS tasks.Source: Table III · p. 7, A. Results of IGD and MSS · p. 7

B. Convergence Curves

  • MO-MFEA converges to both Pareto fronts faster than NSGA-II on most test problems, although NSGA-II can be slightly faster early before MO-MFEA catches up.The early-stage reversal is particularly clear on CILS.Source: B. Convergence Curves · p. 7, Figure 2 · p. 8

A. Results of IGD and MSS

  • MO-MFEA achieves better overall performance than NSGA-II on every proposed test problem except NILS, based on average MSS differences.All reported differences are below zero except for NILS.Source: A. Results of IGD and MSS · p. 7, Table IV · p. 8

C. Distribution of Solutions in the Objective Space

  • MO-MFEA achieves a better Pareto-front approximation than NSGA-II on almost all considered tasks in representative runs selected near average IGD.The distributions show nondominated objective vectors for each task; the selected run is closest to average IGD.Source: C. Distribution of Solutions in the… · pp. 7–8
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