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 GRADUATE SCHOOL SEMINAR

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. 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, instead a representative set of well distributed Pareto solutions will suffice. 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 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), 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). 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.

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, The non-dominated set from EMOA is now clustered and a representative set is constructed choosing representative points from each cluster. Pseudo-weight vector (Deb et. al. (2001)) is calculated for the representative set. This preference information is used by means of changing the weights in ASFs. 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. Pareto-Point

Initial Studies Two objective test problems (ZDT1,ZDT2 & ZDT3) have been chosen.

Future Work & Conclusion 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 Finally, Initial Results are promising and methodology has the capacity to grow into strong rooted procedure to solve Multi-objective problems.

References Deb, K.(2001). Multi-Objective Optimization Using Evolutionary Algorithms. John Wiley & Sons, Chichester. Meittinen, K. (1999). Nonlinear Multiobjective Optimization. Boston: Kluwer. Srinivas, N. and Deb, K. (1994). Multi-objective function optimization using non-dominated sorting genetic algorithms. Evolutionary Computing Journal 2(3), Tamura, H.,Shibata, T., Tomiyama, S., Hatono, I. (1999), A Meta-Heuristic Satisficing Tradeoff Method for Solving Multiobjective Combinatorial Optimization Problems- With Application to Flowshop Scheduling, In Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, vol. 3, pp Deb, K., Sundar, J. (2006). Reference point based multi-objective optimization using evolutionary algorithms. In Proceedings of the Genetic and Evolutionary Computation Conference, pp Deb, K., Kumar, A. (2007). Interactive evolutionary multi-objective optimization and decision making using reference direction method. In Proceedings of the Genetic and Evolutionary Computation Conference, pp Ishibuchi, H., Murata, T. (1998), Multi-objective Genetic Local search Algorithm and Its Application to Flowshop Scheduling. IEEE Transactions on Systems, Man and Cubernetics, 28, 3, Jaszkiewicz, A. (to appear), Genetic local search for multi-objective combinatorial optimization. European Journal of Operation Research. Ishibuchi, H., Doi, T., Nojima, Y.(2006). Incorporation of scalarizing fitness functions into evolutionary multiobjective optimization algorithms, Lecture Notes in Computer Science 4193: Parallel Problem Solving from Nature - PPSN IX, pp Meittinen, K., Mäkelä, MM. (2002). On scalarizing fucntions in multiobjective optimization. OR spectrum, pp Luque, M., Meittinen, K., Eskelinen, P., Ruiz, F. (2007). Incorporating preference information in interactive refernce point methods for multiobjective optimization. Omega, (To appear). Deb, K., Goel, t. (2001). A hybrid multi-objective evolutionary approach to engineering shape design. In Proceedings of the Third International Conference on Evolutionary Multi-criterion Optimization (EMO-2001), pp