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    • Guaranteed Convergence and Distribution in Evolutionary Multi-Objective Algorithms (EMOA’s) via Achivement Scalarizing Functions By Karthik Sindhya a Thesis Supervisors Prof. Kalyanmoy Deb a Prof. Kaisa Miettinen b a Kanpur Genetic Algorithms Laboratory, IIT Kanpur b Quantitative Methods in Economics, HSE Helsinki STATE OF ART SEMINAR
    • Overview
      • Introduction
      • Motivation
      • Literature survey
      • Proposed Methodology
      • Initial Studies
      • Proposed Plan of Research
      • Conclusion
    • Introduction
      • Real world applications have multiple conflicting objectives.
      • Genetic Algorithms (GA’s) work on a population of solutions, a number of Pareto-optimal solutions can be captured in one single run of a multi-objective GA.
      • This makes them naturally suited to solving multi-objective optimization problems.
      • There are two goals in a multi-objective optimization:
        • Find a set of solutions as close as possible to the Pareto-optimal front.
        • And find a set of solutions as diverse as possible .
      • There are two communities dealing with multi-objective optimization problems:
        • MCDM (Multiple Criteria Decision Making) and
        • EMO (Evolutionary Multi-objective optimization).
    • Introduction (Cont’d)
      • The sister field to EMO, MCDM has many variants to generate Pareto optimal solutions.
        • Interactive methods, which involves continuous information flow between decision maker(DM) and the system, are commonly used.
          • Reference Point methods - most widely used class of interactive procedures for multi-objective problems.
      • Reference points formed by aspiration values of the DM.
        • The reference point used to derive achievement scalarizing functions (ASF) which have minimal solutions at Pareto optimal points (ONLY).
      • The importance with the ASF is that the decision maker can obtain any arbitrary weakly Pareto optimal or Pareto optimal solution by moving the reference point only.
      • MOEAs with MCDM prove advantageous and intial work on such combinations are already underway.
    • Motivation
      • None of the EMOA’s guarentee to identify optimal trade-offs in finite number of generations even for simple problems.
      • They can only generate a set of solutions whose objective vectors are hopefully not too far away from the optimal objective vectors.
        • A serious set back, considering the computational expensive nature of real world optimization problems.
      • Main goal in EMO is to generate only representative set of Pareto solutions for Decision maker (DM), so that, (s)he can get an idea of different trade-offs.
      • Hence it makes less sense to get a large number of points to represent pareto front, as most of EMOAs operate present day .
    • Motivation (Cont’d)
      • Best way - Generate a representative set of well distributed pareto solutions whose cardinality is much less than the intial population of EMOA.
        • Savings on function evaluations.
      • Depending on the DMs choice, further solutions in the vicinity can be explored.
      • Innovization , an important post optimal analysis, is meaningless on hopefully near optimal objective vectors.
      • Main Goals of this study -
        • Provide EMO convergence property
        • While keeping diversity within entire or on partial Pareto optimal set.
        • Achieve both tasks in computationally efficient way.
    • Literature Survey
      • Several authors have proposed multi-objective metaheuristics procedures:
        • Schaffer (1985), Fonseca and Flemming (1993), Horn, Nafpliotis and Goldberg(1994), Srinivas and Deb (1995) are based on EA.
        • Serafini (1994), Czyzak and Jaszkiewicz(1998), Ulungu et al. (1999) based on Simulated Annealing (SA).
        • Gandibleux et al. (1996) and Hansen(1998) are based on Tabu Search.
      • Literature in the direction of fostering synergy between EMO and MCDM communities has yeilded differernt approaches:
        • Find a preferred solution to the decision maker from a set of non-dominated points generated by EMOA
        • Incorporating preference information in EMOA.
        • STOM in EMO by Tamura (1999), Reference point based EMO by Deb et al. (2006), Interactive EMO and decision making using reference diraction by Deb et al. (2007), A preference based interactive EMO by Thiele et al. (2007) etc.,
    • Literature Survey (Cont’d)
      • First step in the direction of hybridizing EA with scalarizing fitness function to generate approximately efficient solutions was considered by Ishibuchi et al. (1998).
      • Their idea was followed up by Jaszkiewicz (2002) and he proposed multi-objective genetic local search (MOGLS).
        • Idea is simultaneous optimization of all randomly selected weighted linear utility functions.
        • A single iteration of MOGLS consists of single recombination of a pair of solutions.
        • Offspring is used as a starting point for local search.
      • Above algorithm can be considered as an competitor to our approach.
      • Ishibuchi et. al.(2006), also proposed an idea of integrating Scalarizing Fitness Functions into EMO algorithms.
      • The idea is to probabilistically using a scalarizing fitness function (weighted sum fitness fucntion) for parent selection and generation update in EMO algorithms.
        • The results quoted show improved performance of hybrid EMOA as compared to classical EMOAs.
    • Literature Survey (Cont’d)
      • In MCDM community many kinds of scalarizing functions have been suggested:
        • Ruiz et.al., analyze nine weighing schemes which are used in achievement scalarizing functions.
        • Miettinen et. Al. (2002) also compare their theoretical properties and behavior.
      • Mariano et al., (2007) - Introduced new ways of utilizing preference information specified by the decision maker in interactive reference point based methods.
    • Proposed Methodology
      • In contemporary stage, methodology executes as a serial process and involves following stages:
        • Execute EMOA to get a non-dominated set which is near Pareto-optimal,
          • If this set of points is far from the Pareto optimal set, we get a wider description of Pareto optimal set and finer description if the set is near Pareto optimal. So, a balance of these two scenarios is pre-requisite.
        • The non-dominated set from EMOA is now clustered and a representative set is constructed choosing representative points from each cluster.
          • Note that the extreme points in the non-dominated set is not clustered and is necessarily included into the representative set.
    • Proposed Methodology (Cont’d)
        • Pseudo-weight vector (Deb et. al. (2001)) is calculated for the representative set.
          • Equation below, calculates relative distance of each function value from worst value.
          • They indicates relative trade-off value (preference information) between objectives for all obtained non-dominated solutions.
      Pareto-Point
    • Proposed Methodology (Cont’d)
      • Mariano et al. (2007) introduced new ways of utilizing preference information specified by the decision maker.
        • The goal here was to reflect the preference information by means of changing the weights in ASFs.
        • It can noted here that as we move from one extreme point to another our preference information helps the ASF to generate a well distributed Pareto-optimal set.
      • The above procedure cannot guarantee the extreme points of the Pareto-optimal set.
        • Above problem is handled by generating reference points by perturbing the extreme points in the representative set.
        • Procedure is stopped when we do not make any improvements in extreme solutions in successive iterations.
    • Initial Studies
      • Two objective test problems (ZDT1,ZDT2 & ZDT3) have been chosen.
        • ZDT1: This is a 30 variable problem having convex Pareto front.
          • NSGA-II : 50 generations
          • NSGA Pop size : 100..
          • Clustered Points: 11
          • RGA : 5000
          • RGA pop size: 200
    • Initial Studies (Cont’d)
      • ZDT2: This is also a 30 variable problem having non-convex Pareto-optimal front.
          • NSGA-II : 75 generations
          • NSGA Pop size : 100.
          • Clustered Points: 11
          • RGA : 5000
          • RGA pop size: 200
    • Initial Studies (Cont’d)
      • ZDT3: This is an 30 variable problem having a number of disconnected Pareto-optimal fronts.
          • NSGA-II : 75 generations.
          • NSGA Pop size : 100
          • Clustered Points: 11
          • RGA : 5000
          • RGA pop size: 200
    • Proposed Plan for Research
      • Investigate above approach with
        • More objectives
        • Interactive applications
      • Compare with MOGLS
        • Computational time & accuracy
      • Investigate the procedure with other scalarizing functions
      • Implement concurrent integration
        • Local search as a special operator within EMOA
      • Application to engineering problems
    • Conclusion
      • Initial Results are promising and methodology has the capacity to grow into strong rooted procedure to solve Multi-objective problems.
    • Acknowledgement
      • I would acknowledge the support given by Academy of Finland and HSE
    •