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  1. 1. 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
  2. 2. Overview <ul><li>Introduction </li></ul><ul><li>Motivation </li></ul><ul><li>Literature survey </li></ul><ul><li>Proposed Methodology </li></ul><ul><li>Initial Studies </li></ul><ul><li>Proposed Plan of Research </li></ul><ul><li>Conclusion </li></ul>
  3. 3. Introduction <ul><li>Real world applications have multiple conflicting objectives. </li></ul><ul><li>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. </li></ul><ul><li>This makes them naturally suited to solving multi-objective optimization problems. </li></ul><ul><li>There are two goals in a multi-objective optimization: </li></ul><ul><ul><li>Find a set of solutions as close as possible to the Pareto-optimal front. </li></ul></ul><ul><ul><li>And find a set of solutions as diverse as possible . </li></ul></ul><ul><li>There are two communities dealing with multi-objective optimization problems: </li></ul><ul><ul><li>MCDM (Multiple Criteria Decision Making) and </li></ul></ul><ul><ul><li>EMO (Evolutionary Multi-objective optimization). </li></ul></ul>
  4. 4. Introduction (Cont’d) <ul><li>The sister field to EMO, MCDM has many variants to generate Pareto optimal solutions. </li></ul><ul><ul><li>Interactive methods, which involves continuous information flow between decision maker(DM) and the system, are commonly used. </li></ul></ul><ul><ul><ul><li>Reference Point methods - most widely used class of interactive procedures for multi-objective problems. </li></ul></ul></ul><ul><li>Reference points formed by aspiration values of the DM. </li></ul><ul><ul><li>The reference point used to derive achievement scalarizing functions (ASF) which have minimal solutions at Pareto optimal points (ONLY). </li></ul></ul><ul><li>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. </li></ul><ul><li>MOEAs with MCDM prove advantageous and intial work on such combinations are already underway. </li></ul>
  5. 5. Motivation <ul><li>None of the EMOA’s guarentee to identify optimal trade-offs in finite number of generations even for simple problems. </li></ul><ul><li>They can only generate a set of solutions whose objective vectors are hopefully not too far away from the optimal objective vectors. </li></ul><ul><ul><li>A serious set back, considering the computational expensive nature of real world optimization problems. </li></ul></ul><ul><li>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. </li></ul><ul><li>Hence it makes less sense to get a large number of points to represent pareto front, as most of EMOAs operate present day . </li></ul>
  6. 6. Motivation (Cont’d) <ul><li>Best way - Generate a representative set of well distributed pareto solutions whose cardinality is much less than the intial population of EMOA. </li></ul><ul><ul><li>Savings on function evaluations. </li></ul></ul><ul><li>Depending on the DMs choice, further solutions in the vicinity can be explored. </li></ul><ul><li>Innovization , an important post optimal analysis, is meaningless on hopefully near optimal objective vectors. </li></ul><ul><li>Main Goals of this study - </li></ul><ul><ul><li>Provide EMO convergence property </li></ul></ul><ul><ul><li>While keeping diversity within entire or on partial Pareto optimal set. </li></ul></ul><ul><ul><li>Achieve both tasks in computationally efficient way. </li></ul></ul>
  7. 7. Literature Survey <ul><li>Several authors have proposed multi-objective metaheuristics procedures: </li></ul><ul><ul><li>Schaffer (1985), Fonseca and Flemming (1993), Horn, Nafpliotis and Goldberg(1994), Srinivas and Deb (1995) are based on EA. </li></ul></ul><ul><ul><li>Serafini (1994), Czyzak and Jaszkiewicz(1998), Ulungu et al. (1999) based on Simulated Annealing (SA). </li></ul></ul><ul><ul><li>Gandibleux et al. (1996) and Hansen(1998) are based on Tabu Search. </li></ul></ul><ul><li>Literature in the direction of fostering synergy between EMO and MCDM communities has yeilded differernt approaches: </li></ul><ul><ul><li>Find a preferred solution to the decision maker from a set of non-dominated points generated by EMOA </li></ul></ul><ul><ul><li>Incorporating preference information in EMOA. </li></ul></ul><ul><ul><li>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., </li></ul></ul>
  8. 8. Literature Survey (Cont’d) <ul><li>First step in the direction of hybridizing EA with scalarizing fitness function to generate approximately efficient solutions was considered by Ishibuchi et al. (1998). </li></ul><ul><li>Their idea was followed up by Jaszkiewicz (2002) and he proposed multi-objective genetic local search (MOGLS). </li></ul><ul><ul><li>Idea is simultaneous optimization of all randomly selected weighted linear utility functions. </li></ul></ul><ul><ul><li>A single iteration of MOGLS consists of single recombination of a pair of solutions. </li></ul></ul><ul><ul><li>Offspring is used as a starting point for local search. </li></ul></ul><ul><li>Above algorithm can be considered as an competitor to our approach. </li></ul><ul><li>Ishibuchi et. al.(2006), also proposed an idea of integrating Scalarizing Fitness Functions into EMO algorithms. </li></ul><ul><li>The idea is to probabilistically using a scalarizing fitness function (weighted sum fitness fucntion) for parent selection and generation update in EMO algorithms. </li></ul><ul><ul><li>The results quoted show improved performance of hybrid EMOA as compared to classical EMOAs. </li></ul></ul>
  9. 9. Literature Survey (Cont’d) <ul><li>In MCDM community many kinds of scalarizing functions have been suggested: </li></ul><ul><ul><li>Ruiz, analyze nine weighing schemes which are used in achievement scalarizing functions. </li></ul></ul><ul><ul><li>Miettinen et. Al. (2002) also compare their theoretical properties and behavior. </li></ul></ul><ul><li>Mariano et al., (2007) - Introduced new ways of utilizing preference information specified by the decision maker in interactive reference point based methods. </li></ul>
  10. 10. Proposed Methodology <ul><li>In contemporary stage, methodology executes as a serial process and involves following stages: </li></ul><ul><ul><li>Execute EMOA to get a non-dominated set which is near Pareto-optimal, </li></ul></ul><ul><ul><ul><li>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. </li></ul></ul></ul><ul><ul><li>The non-dominated set from EMOA is now clustered and a representative set is constructed choosing representative points from each cluster. </li></ul></ul><ul><ul><ul><li>Note that the extreme points in the non-dominated set is not clustered and is necessarily included into the representative set. </li></ul></ul></ul>
  11. 11. Proposed Methodology (Cont’d) <ul><ul><li>Pseudo-weight vector (Deb et. al. (2001)) is calculated for the representative set. </li></ul></ul><ul><ul><ul><li>Equation below, calculates relative distance of each function value from worst value. </li></ul></ul></ul><ul><ul><ul><li>They indicates relative trade-off value (preference information) between objectives for all obtained non-dominated solutions. </li></ul></ul></ul>Pareto-Point
  12. 12. Proposed Methodology (Cont’d) <ul><li>Mariano et al. (2007) introduced new ways of utilizing preference information specified by the decision maker. </li></ul><ul><ul><li>The goal here was to reflect the preference information by means of changing the weights in ASFs. </li></ul></ul><ul><ul><li>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. </li></ul></ul><ul><li>The above procedure cannot guarantee the extreme points of the Pareto-optimal set. </li></ul><ul><ul><li>Above problem is handled by generating reference points by perturbing the extreme points in the representative set. </li></ul></ul><ul><ul><li>Procedure is stopped when we do not make any improvements in extreme solutions in successive iterations. </li></ul></ul>
  13. 13. Initial Studies <ul><li>Two objective test problems (ZDT1,ZDT2 & ZDT3) have been chosen. </li></ul><ul><ul><li>ZDT1: This is a 30 variable problem having convex Pareto front. </li></ul></ul><ul><ul><ul><li>NSGA-II : 50 generations </li></ul></ul></ul><ul><ul><ul><li>NSGA Pop size : 100.. </li></ul></ul></ul><ul><ul><ul><li>Clustered Points: 11 </li></ul></ul></ul><ul><ul><ul><li>RGA : 5000 </li></ul></ul></ul><ul><ul><ul><li>RGA pop size: 200 </li></ul></ul></ul>
  14. 14. Initial Studies (Cont’d) <ul><li>ZDT2: This is also a 30 variable problem having non-convex Pareto-optimal front. </li></ul><ul><ul><ul><li>NSGA-II : 75 generations </li></ul></ul></ul><ul><ul><ul><li>NSGA Pop size : 100. </li></ul></ul></ul><ul><ul><ul><li>Clustered Points: 11 </li></ul></ul></ul><ul><ul><ul><li>RGA : 5000 </li></ul></ul></ul><ul><ul><ul><li>RGA pop size: 200 </li></ul></ul></ul>
  15. 15. Initial Studies (Cont’d) <ul><li>ZDT3: This is an 30 variable problem having a number of disconnected Pareto-optimal fronts. </li></ul><ul><ul><ul><li>NSGA-II : 75 generations. </li></ul></ul></ul><ul><ul><ul><li>NSGA Pop size : 100 </li></ul></ul></ul><ul><ul><ul><li>Clustered Points: 11 </li></ul></ul></ul><ul><ul><ul><li>RGA : 5000 </li></ul></ul></ul><ul><ul><ul><li>RGA pop size: 200 </li></ul></ul></ul>
  16. 16. Proposed Plan for Research <ul><li>Investigate above approach with </li></ul><ul><ul><li>More objectives </li></ul></ul><ul><ul><li>Interactive applications </li></ul></ul><ul><li>Compare with MOGLS </li></ul><ul><ul><li>Computational time & accuracy </li></ul></ul><ul><li>Investigate the procedure with other scalarizing functions </li></ul><ul><li>Implement concurrent integration </li></ul><ul><ul><li>Local search as a special operator within EMOA </li></ul></ul><ul><li>Application to engineering problems </li></ul>
  17. 17. Conclusion <ul><li>Initial Results are promising and methodology has the capacity to grow into strong rooted procedure to solve Multi-objective problems. </li></ul>
  18. 18. Acknowledgement <ul><li>I would acknowledge the support given by Academy of Finland and HSE </li></ul>