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  • Enhancing Clearing-based Niching Method UsingDelaunay Triangulation

    Shivam Kalra, Shahryar Rahnamayan, SMIEEE and Kalyanmoy Deb, FIEEEDepartment of System Design Engineering

    University of Waterloo, CanadaEmail: shivam.kalra@uwaterloo.ca

    Department of Electrical, Computer and Software EngineeringUniversity of Ontario Institute of Technology, Canada

    Email: shahryar.rahnamayan@uoit.caDepartment of Electrical and Computer Engineering

    Michigan State University, USAEmail: kdeb@msu.com

    plain [L]978-1-5090-4601-0/17/$31.00 c2017 IEEEAbstractThe interest in multi-modal optimization methods is

    increasing in the recent years since many of real-world optimiza-tion problems have multiple/many optima and decision makersprefer to find all of them. Multiple global/local peaks createdifficulties for optimization algorithms. In this context, nichingis well-known and widely used technique for finding multiplesolutions in multi-modal optimization. One commonly used nich-ing technique in evolutionary algorithms is the Clearing method.However, canonical clearing scheme reduces the explorationcapacity of the evolutionary algorithms. In this paper, DelaunayTriangulation based Clearing (DT-Clearing) procedure is pro-posed to handle multi-modal optimizations more efficiently whilepreserving simplicity of canonical clearing approach. In DT-Clearing, cleared individuals are reallocated in the biggest emptyspaces formed within the search space which are determinedthrough Delaunay Triangulation. The reallocation of clearedindividuals discourages wasting of the resources and allows betterexploration of the landscape. The algorithm also uses an externalmemory, an archive of the explored niches, thus preventingthe redundant visiting of the individuals, henceforth findingmore solutions in lesser number of generations. The methodis tested using multi-modal benchmark problems proposed forthe IEEE CEC 2013, Special Session on Niching Methods forMultimodal Optimization. Our method obtains promising resultsin comparison with the canonical clearing and demonstrates tobe a competitive niching algorithm.

    I. INTRODUCTION

    Evolutionary Algorithms (EAs) are typically devised forconverging to a single solution because of the globally em-ployed selection scheme. However, most of the real-worldproblems exhibit the property of having more than onesatisfactory solution. Process that involves finding multipleviable solutions for a given optimization problem is calledmulti-modal optimization. In particular, such problems are ofgreat abundance in science and engineering viz. aerodynamicdesign, construction, scheduling, time/cost/resource schedul-ing. Multiple solutions for a given optimization problems areconceptualized as peaks (or troughs if minimization problemis assumed) and are often referred to as niches.

    It is usually desirable to find multiple feasible solutionsas it enables decision makers or experts to select the mostappropriate solution depending of the domain and constraintsof the problem. Inspired by the ecosystems in nature, nichingmethods are commonly employed for tasks dealing with multi-modal optimization. Niching is a generic term referred to astechnique of locating and preserving the stable peaks/niches, orany potential candidates for optimum during the EA process.

    All ecosystems have many different physical spaces (niches)with a finite amount of resources, which are apt for differentinter-competing species. For example, on Earth, organisms liv-ing on land have different characteristics than organisms livingin water bodies, allowing each of these groups of organisms toevolve independently within their respective niches. Thus, theecosystem encourages the diversity and allows the preservationof various dissimilar species within their respective niches.

    In context of optimization algorithms, niching methods arealso inspired by nature, enabling to split the population intodistinct sub-populations (niches) searching certain areas of thesearch space [1]. A niching method can be embedded into astandard EA to promote and maintain formation of multiplestable sub-populations within a single population, with thegoal of finding multiple globally optimal or sub-optimal solu-tions. In a scenario of EA, the fitness symbolizes the resourcesof the niches and species are individuals grouped accordingto certain criteria (vicinity, fitness value and etc.), while nichecorresponds to an optimum of the fitness landscape.

    Evolutionary algorithms are well-established as strong can-didates for tackling uni-modal optimization problems. How-ever in multi-modal domain, many challenges exist; for in-stance, most niching techniques are not efficient in solvinga multi-modal problem of a relatively large scale or withlarge number of optima, or detection of niches with variableradius or peak values. In addition, drastic limitations on theircomputation complexity still persist. Therefore, this field isan active area of research concerning development of nichingstrategies for EA as to benefit from their synergy for efficientlyhandling multi-modality.

    978-1-5090-4601-0/17/$31.00 c2017 IEEE2328

  • Several niching methods have been developed previously,such as, stretching and deflation [2], crowding [3], fitnesssharing [4], deterministic crowding [5], restricted tournamentselection [6], clearing [7] etc. The comparisons have shownthat clearing methods are efficient in reducing the geneticdrift and maintaining multiple stable solutions [8]. Howeverin the canonical clearing approach, cleared individuals haveno chance to participate in the mutation and crossover, whichlimits the exploration capabilities of evolutionary process.

    In multi-modal optimization domain, two criteria are gen-erally used to measure the success of the search algorithms.First, whether an optimization algorithm can find all desiredglobal/local optima within reasonable amount of time, and thesecond, if is capable of stably maintaining multiple candidatesolutions. Clearing method for niching is successful in achiev-ing the latter, however it falls short in former criterion. We areinterested not only in stably identifying one or more globaloptima but we are interested to locate set of all acceptablesolutions in timely manner.

    In this paper, we present a novel multi-modal optimizationalgorithm based on an enhanced clearing procedure, we callit Delaunay Triangulation Based Clearing (DT-Clearing). Inthis algorithm, cleared individuals are reallocated in the centerof the large empty hyper-spheres formed within the searchspace. The proposed algorithm uses Delaunay Triangulationtechnique to find the large empty hyper-sphere. The pro-posed approach allows the non-elitist search by reallocationof cleared individuals, it is able to form stable niches acrossdifferent local neighborhoods and eventually locates multipleglobal/local optima in lesser number of generations whencompared with canonical clearing method.

    The remainder of this paper is organized as follows. Sec-tion II provides brief problem statement for multi-modal op-timization. Section III presents background and related work.Detailed explanation of algorithm is presented in Section IVand Section V covers results of our algorithm against bench-mark test suite proposed for the IEEE CEC 2013, SpecialSession on Niching Methods for Multimodal Optimizationand comparison with canonical clearing method and modifiedclearing approach. The conclusion remarks are presented inSection VI.

    II. PROBLEM STATEMENT

    The general aim of multi-modal optimization is similar tothat of the standard optimization task, that is, in a given searchdomain X , we seek to maximize/minimize

    f(x), x X (1)

    For this paper, X is subjected to box-constraint, ie. any givenx X follows

    xl x xu (2)

    Where xl and xu are lower and upper bounds of x.In the case of multi-modal optimization, we seek to find

    multiple x X that satisfies the constraint in Eq 2 andattains the maximum possible value of f(x) in vicinity of x

    (local maxima). Therefore a given multi-modal optimizationalgorithms must successfully identify and maintain all the x.

    III. NICHING METHODS: A BRIEF REVIEW

    Simple Genetic Algorithm (SGA) is designed to convergeat single solution; therefore in their classical form they arenot useful in context of solving multi-modal problems. Thislimitation of SGA can be overcome by a mechanism of nichingwhich allows GA process to identify multiple solutions. Thesingle convergence characteristic of GA occurs because ofthe Genetic Drift which is an artifact of stochastic selectionprocess used in GA with finite chromosome in the population.A naive niching method would be running SGA several timeson the same problem and due to the stochastic nature of GA,one could attain multiple different solutions. However, mostniching methods involve modifying the GA operators to allowformation and identification of multiple niches/solutions. Oneof the earliest niching method is Cavicchios Pre-selection Op-erator [9] which was generalized to Crowding by De Jong [10]whereby the individuals produced by crossover and mutationof parents, replaces the most similar parent if it has a betterfitness. Mengshoel in [5] made modifications to Crowding toreduce replacement errors, restore selection pressure, and re-move the parameter called crowding factor (CF), thus resultingin the Probabilistic Crowding.

    Another popular niching method is Fitness Sharing [4];whereby fitness of an individual is lowered if there aremany other similar individuals similar to it thus forcing GAto maintain diversity within the population. Fitness sharingtechnique has several drawbacks, e.g. it depends on the valuesof two parameters (the niche radius and the scaling factor),which cannot be easily determined. As a consequence, moreadvanced fitness sharing methods have been cr

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