NAEMO: Neighborhood-sensitive archived evolutionary many-objective optimization algorithm

https://doi.org/10.1016/j.swevo.2018.12.002Get rights and content

Highlights

  • Proposed approach (NAEMO) establishes and explores the neighborhood property

  • Presents probabilistic mutation switching scheme for improved candidate generation

  • Demonstrates monotonic improvement in diversity, theoretically and experimentally

  • Theoretically analyzes penalty factor of PBI function used in decomposition step

  • Shows superiority on DTLZ problems and M2M-like performance on some IMB problems

Abstract

One of the prominent strategies to address many-objective optimization problems involves using the reference direction based algorithms. However, literature severely lacks formal mathematical analysis to establish the reason behind superior performance of such methods. In this work, the neighborhood property of the many-objective optimization problems is recognized and is used to propose the neighborhood-sensitive archived evolutionary many-objective optimization (NAEMO) algorithm. In NAEMO, mating occurs within a local neighborhood and every reference direction continues to retain at least one associated candidate solution. Such preservation of candidate solutions leads to a monotonic improvement in diversity which has been theoretically and experimentally studied. Moreover, to combine the advantages of various mutation strategies, probabilistic mutation switching concept is introduced and to keep the archive size under control, periodic filtering modules are integrated with the NAEMO framework. Experimental results reveal that, in terms of inverted generational distance, hypervolume values and purity metric, NAEMO outperforms several state-of-the-art algorithms viz. NSGA-III, MOEA/D, θ-DEA, MOEA/DD, GrEA, HypE, MOPSO and dMOPSO on DTLZ1-4 test problems for up to 15 objectives. Further experiments show that NAEMO is competitive to M2M-based algorithms where the difficult regions of IMB problems have also been explored. These experiments make NAEMO a robust algorithm, which is additionally supported by theoretical foundations. The source code of NAEMO is available at http://worksupplements.droppages.com/naemo.

Introduction

Many-Objective Optimization (MaOO) algorithms are used in almost all application domains and hence, developments in this area have always been open to the researchers. Evolutionary algorithms are popular as these are capable of handling a population of solutions rather than a single solution in every iteration and also of finding approximate optimal solutions even for hard problems [1]. Many-objective evolutionary algorithms (MaOEAs) are used to address optimization problems with box constraints as defined in Eq. (1), where d-dimensional candidate vectors (X=x1,,xd) get mapped to M-dimensional objective vectors (F(X)). Formally, when M ≥ 4, the particular sub-class is called many-objective optimization problem [[1], [2], [3], [4]]. Applications of MaOO algorithms are found in control systems [5,6], brain-computer interfacing [7], in bioactive compound extraction [8], in box-pushing problem of robotics [9], in structural optimization of shed truss [10] and many more areas.MinimizeF(X)=f1(X),,fM(X)where,XRd,F(X):RdRMandxjLxjxjU,j=1,2,,d

Among other approaches, reference line based algorithms [11] such as non-dominated sorting genetic algorithm – III (NSGA-III) [12], θ-dominance based evolutionary algorithm (θ-DEA) [13], decomposition based multi-objective evolutionary algorithm (MOEA/D) [14], dominance and decomposition based multi-objective evolutionary algorithm (MOEA/DD) [15] and several other variants and extensions of these algorithms have been developed and shown to perform well for problems with number of objectives as high as 15. MOEA/D introduced the concept of using reference lines effectively and presented extremely promising results [1,3]. Due to the nature of the algorithm which basically decomposes a multi-objective optimization (MOO) into several single-objective optimization problems, it has lent itself to several modified and extended algorithms as developed in Refs. [15,16]. NSGA-III combined the concept of non-dominated sorting with a diversity maintenance operator based on reference lines unlike in NSGA-II [17] which uses a crowding distance operator. The θ-DEA algorithm introduced the concept of θ-dominated sorting which combines the penalty based boundary intersection (PBI) function value with non-dominated sorting. MOEA/DD has effectively combined the concepts of dominance as well as decomposition and presented the state-of-the-art results. The concept of decomposition in MOEA/D has also been combined with other meta-heuristics such as PSO yielding dMOPSO [18].

Recently, a new set of test problems referred to as Imbalanced problems [19] have been introduced. These problems have ‘difficult’ regions which create difficulty for the general algorithms to obtain the complete Pareto-front. Following this, the M2M based algorithms have been introduced [19,20] which decomposes a multi-objective problem into several sub-multi-objective optimization problems. The paper theoretically proves that the M2M strategy would yield better results.

Theoretical analyses and results are extremely necessary for understanding optimization problems and algorithms. The work in Ref. [21] shows that theoretical results pertaining to single objective optimization do not carry over to the multi-objective case. It finally develops an algorithm based on the theory which converges with probability 1 for a certain test function. The work in Ref. [22] performs theoretical analyses on the convergence of multi-objective evolutionary algorithms. The study in Ref. [23] presents theoretical analyses of decomposition based multi-objective optimization algorithms. However, much theoretical work on MaOO is still not present in the literature. The working and reasoning behind the performance of the proposed algorithms are also usually qualitative. Formal theoretical analysis will aid in finding the weaknesses of algorithms as well as making improvements with a concrete theoretical basis. This paper has been written as an effort towards filling this gap. The motivations and contributions of this paper are enlisted below:

  • 1.

    This paper presents the neighborhood property which is followed by a many-objective optimization algorithm along with its theoretical and experimental studies.

  • 2.

    The Penalty-based Boundary Intersection (PBI) function [14] has gained popularity and is used very widely by MaOO algorithms, especially in ones which are an extension of MOEA/D. A theoretical analysis has been presented on the PBI function and how the shape of the actual Pareto-front affects the final solution if the PBI function is used.

  • 3.

    Based on these theoretical concepts, we propose a novel many-objective optimization algorithm viz., Neighborhood-sensitive Archived Evolutionary Many-objective Optimization Algorithm (NAEMO) which has been shown to significantly outperform other state-of-the-art algorithms in majority of the cases on the DTLZ test suite.

  • 4.

    NAEMO introduces convergence-based filtering and diversity-based filtering schemes, followed by a theoretical analysis of these two operations.

  • 5.

    Since the proposed algorithm guarantees diversity preservation, as proven in the later sections, this algorithm has also been evaluated on the imbalanced test problems and shown to have competitive performance with the previous state-of-the-art, multiobjective-to-multiobjective (M2M)-based algorithms [19] on the IMB test suite.

  • 6.

    NAEMO shows a way of using both PBI function and Pareto-dominance simultaneously. As the proposed algorithm is also very modular, it lends itself to easy extensions and modifications.

  • 7.

    Effort has been made towards making a better candidate vector generation scheme by introducing the probabilistic mutation switching concept.

In order to prove the effectiveness of NAEMO, the DTLZ [24] and IMB [19] test suites have been considered. The IMB test suite has been considered owing to the fact that NAEMO guarantees diversity preservation much like the M2M [19] algorithms. NAEMO has been compared to MOEA/DD [15], MOEA/D [14], θ-DEA [13], NSGA-III [12], Hypervolume estimator based evolutionary algorithm (HypE) [25] and grid based evolutionary algorithm (GrEA) [26] on the DTLZ test suites. For the IMB test suite, NAEMO has been compared to the previous state of the art in this test suite - M2M algorithms. Experimental results clearly show that NAEMO outperforms other state-of-the-art algorithms in the DTLZ test suite and by a large margin. It is also interesting to see that NAEMO is successful in obtaining a decent Pareto-front for the imbalanced problems in spite of the difficult regions and even outperforms the M2M algorithms in some of the cases.

The remainder of the paper is structured as follows. Section 2 states the neighborhood theorem and provides a Proof for it along with visualization. Section 3 presents a detailed theoretical analysis of the effect of the shape of the Pareto-front on the performance of the PBI function. Section 4 presents the new algorithm, NAEMO along with preliminary theoretical analyses. This is followed by Section 5 which presents comparison of NAEMO with other state of the art algorithms on DTLZ and IMB test suites. The paper finally ends with the conclusion section.

Section snippets

Theoretical outline of the neighborhood property

A notion of the spatial relationship between objective space and decision space, which is created by partitioning the objective space using reference vector based association of evolutionary candidate solutions, is conveyed by the following theorem based on which a new Many-Objective Optimization algorithm is proposed.

Theorem 1

(Neighborhood property) The regions corresponding to each reference line in the objective space which share a common boundary also share a common boundary in the decision space

Analysing the penalty factor of the penalty-based boundary intersection approach

Penalty-based Boundary Intersection (PBI) function [14] is used in several multi-objective optimization algorithms and has proven its efficacy. It helps in decomposing a multi-objective problem into several sub-problems and provides a measure of fitness for each of the sub-problems. The PBI function is a value that is calculated for a point and for a given reference line as shown in Eq. (2).g=d1+θd2where, d1 is the magnitude of the projection of the given point on the given reference line in

Neighborhood-sensitive archived evolutionary many-objective optimization (NAEMO): the proposed approach

In this section, we discuss in detail different steps of the proposed Neighborhood-sensitive Archived Evolutionary Many-objective Optimization (NAEMO) and the underlying features of these mechanisms.

The key concepts used in NAEMO are as follows:

  • 1.

    Using the neighborhood property: NAEMO maintains a very organized archive to store the population. The global archive is divided into sub-archives corresponding to each reference line. Each sub-archive stores population members associated with the

Experimental results and interpretations

In this section, we present a comparison of NAEMO with other state of the art algorithms on DTLZ1-DTLZ4 test functions from the DTLZ test suite [24] and IMB1-IMB9 test functions from the Imbalanced problems test suite [19]. For each DTLZ test function, the number of objectives is set as M ∈ {3, 5, 8, 10, 15}. According to the recommendations of [24], the number of decision variables for a particular DTLZ test function is set as d = M + s − 1, where s = 5 for DTLZ1 and s = 10 for DTLZ2, DTLZ3

Conclusion and future research scope

Motivated by the success of the reference direction-based evolutionary many-objective optimization algorithms, this work proposes a novel approach viz. Neighborhood-sensitive Archived Evolutionary Many-objective Optimization (NAEMO) where the neighborhood property of the many-objective optimization problems is identified and used for selecting the mating candidate solutions for generation of new candidate solutions. Moreover, NAEMO aims to preserve and monotonically improve the diversity

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