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A Parallel Version of SMS-EMOA for Many-Objective Optimization Problems

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 9921)


In the last decade, there has been a growing interest in multi-objective evolutionary algorithms that use performance indicators to guide the search. A simple and effective one is the \(\mathcal {S}\)-Metric Selection Evolutionary Multi-Objective Algorithm (SMS-EMOA), which is based on the hypervolume indicator. Even though the maximization of the hypervolume is equivalent to achieving Pareto optimality, its computational cost increases exponentially with the number of objectives, which severely limits its applicability to many-objective optimization problems. In this paper, we present a parallel version of SMS-EMOA, where the execution time is reduced through an asynchronous island model with micro-populations, and diversity is preserved by external archives that are pruned to a fixed size employing a recently created technique based on the Parallel-Coordinates graph. The proposed approach, called \(\mathcal {S}\)-PAMICRO (PArallel MICRo Optimizer based on the \(\mathcal {S}\) metric), is compared to the original SMS-EMOA and another state-of-the-art algorithm (HypE) on the WFG test problems using up to 10 objectives. Our experimental results show that \(\mathcal {S}\)-PAMICRO is a promising alternative that can solve many-objective optimization problems at an affordable computational cost.


These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

C.A. Coello Coello—Author gratefully acknowledges support from CONACyT project no. 221551.

E. Alba—Author is partially funded by the Spanish MINECO and FEDER project TIN2014-57341-R (

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  1. 1.

    A solution \(\varvec{x} \in \mathcal {S}\) dominates a solution \(\varvec{y} \in \mathcal {S}\) (\(\varvec{x} \prec \varvec{y}\)) if and only if \(\forall i \in \left\{ 1,\ldots ,m\right\} \), \(f_{i}(\varvec{x}) \le f_{i}(\varvec{y})\) and \(\exists j \in \left\{ 1,\ldots ,m\right\} \), \(f_{j}(\varvec{x}) < f_{j}(\varvec{y})\).

  2. 2.

    A density estimator models the distribution of a population, by measuring the similarity degree among individuals.

  3. 3.

    Two solutions \(\varvec{x}, \varvec{y} \in \mathcal {S}\) are incomparable if neither \(\varvec{x} \prec \varvec{y}\) nor \(\varvec{y} \prec \varvec{x}\) holds.

  4. 4.

    Diversity refers to achieving a uniform distribution of solutions covering all regions of the objective function space.

  5. 5.


  6. 6.

    A performance indicator, defined as , measures the quality of an approximation set (the final population of a MOEA).

  7. 7.

    This is known as migration.

  8. 8.

    Available at

  9. 9.


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Correspondence to Raquel Hernández Gómez .

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Hernández Gómez, R., Coello Coello, C.A., Alba, E. (2016). A Parallel Version of SMS-EMOA for Many-Objective Optimization Problems. In: Handl, J., Hart, E., Lewis, P., López-Ibáñez, M., Ochoa, G., Paechter, B. (eds) Parallel Problem Solving from Nature – PPSN XIV. PPSN 2016. Lecture Notes in Computer Science(), vol 9921. Springer, Cham.

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