Principles in Noisy Optimization pp 243-305 | Cite as

# Noisy Multi-objective Optimization for Multi-robot Box-Pushing Application

## Abstract

The chapter proposes an extension of multi-objective optimization realized with differential evolution algorithm to handle the effect of noise in objective functions. The proposed extension offers four merits with respect to its traditional counterpart by the following counts. First, an adaptive selection of sample size of objective functions (based on their variance in the neighborhood) avoids unnecessary re-evaluation for quality solutions without avoiding the necessary evaluation for the relatively poor solutions. Second, introduction of a probabilistic dominance scheme (in contrast to traditional deterministic selection) to measure the qualitative dominance of one solution with respect to the other reduces the possibility of promotion of inferior solutions into the Pareto front. Third, the solutions lying in the neighborhood of the apparent Pareto front are clustered with an aim to nullify the detrimental effect of noise on the rejection of the non-dominated solutions from the Pareto-optimal front. Finally, to determine the diversity of solutions in the noisy objective space, the crowding distance metric is modified using the probability of a solution having been dominated by others. Computer simulations performed on noisy version of a well-known set of 23 benchmark functions reveal that the proposed algorithm outperforms its competitors with respect to inverted generational distance, spacing, and error ratio. Experiments undertaken on a multi-objective formulation of a multi-robot cooperative box-pushing problem indicate that the proposed algorithm outperforms other multi-objective optimization techniques used for the same application with respect to three standard metrics defined in the literature.

## Keywords

Noisy multi-objective optimization Differential evolution Pareto co-ranking Probabilistic dominance Multi-robot box-pushing## References

- 1.D. Buche, P. Stall, R. Dornberger, P. Koumoutsakos, Multiobjective evolutionary algorithm for the optimization of noisy combustion processes. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev.
**32**(4), 460–473 (2002)CrossRefGoogle Scholar - 2.J. Branke, C. Schmidt, Selection in the presence of noise in
*Genetic and Evolutionary Computation*. Lecture Notes in Computer Science, Vol. 2723, pp. 766–777 (2003)CrossRefGoogle Scholar - 3.Y. Sano, H. Kita, Optimization of noisy fitness functions by means of genetic algorithms using history of search with test of estimation. Proc. Cong. Evol. Comput.
**1**, 360–365 (2002)Google Scholar - 4.J.M. Fitzpatrick, J.J. Greffenstette, Genetic algorithms in noisy environments. Mach. Learn.
**3**, 101–120 (1994)Google Scholar - 5.P. Stagge, Averaging Efficiently in the Presence of Noise, in
*Parallel Problem Solving from Nature*V. ser. LNCS, vol. 1498, ed. by A.E. Eiben et al. (Springer, Berlin, Germany, 1998), pp. 188–197Google Scholar - 6.A.N. Aizawa, B.W. Wah, Dynamic control of genetic algorithms in a noisy environment, in
*Proceedings of Conference Genetic Algorithms*(1993), pp. 48–55Google Scholar - 7.J. Branke, C. Schmidt, in
*Selection in the Presence of Noise*, ed. by E. Cantu-Paz. Lecture Notes in Computer Science, Proceedings of Genetic Evolutionary Computing Conference, vol. 2723 (2003), pp. 766–777CrossRefGoogle Scholar - 8.B.L. Miller, D.E. Goldberg, Genetic algorithms, selection schemes, and the varying effects of noise. Evol. Comput.
**4**(2), 113–131 (1996)CrossRefGoogle Scholar - 9.L.M. Rattray, J. Shapiro, Noisy fitness evaluation in genetic algorithms and the dynamics of learning, in
*Foundations of Genetic Algorithms*, ed. by R.K. Belew, M.D. Vose (Morgan Kaufmann, San Mateo, CA, 1997), pp. 117–139Google Scholar - 10.H.G. Beyer, Toward a theory of evolution strategies: some asymptotical results from the (1 + λ)-theory. Evol. Comput.
**1**(2), 165–188 (1993)CrossRefGoogle Scholar - 11.B.L. Miller,
*Noise, Sampling, and Efficient Genetic Algorithms*, Ph.D. dissertation, Dept. of Computer Science, Univ. Illinois at Urbana-Champaign, Urbana, IL, available as TR 97001 (1997)Google Scholar - 12.S. Das, A. Konar, U.K. Chakraborty, Improved differential evolution algorithms for handling noisy optimization problems, in
*Proceedings of IEEE Congress of Evolutionary Computation,*vol. 2, pp. 1691–1698 (2005)Google Scholar - 13.S. Markon, D. Arnold, T. Back, T. Beislstein, H.G. Beyer, Thresholding-a selection operator for noisy ES, in
*Proceedings of Congress on Evolutionary Computation*(2001), pp. 465–472Google Scholar - 14.E.J. Hughes, Evolutionary multi-objective ranking with uncertainty and noise, in
*Evolutionary Multi-Criterion Optimization*, EMO, vol. 1993 (2001)Google Scholar - 15.A. Singh,
*Uncertainty based multi-objective optimization of groundwater remediation design*, M.S. thesis, Univ. Illinois at Urbana-Champaign, Urbana, IL, 2003Google Scholar - 16.R. Storn, K.V. Price, Differential evolution–a simple and efficient adaptive scheme for global optimization over continuous spaces (Institute of Company Secretaries of India, Chennai, Tamil Nadu. Tech. Report TR-95-012, 1995)Google Scholar
- 17.J. Teich, Pareto-front exploration with uncertain objectives, in
*Evolutionary Multi-Criterion Optimization*. Lecture Notes in Computer Science, Vol. 1993, (2001), pp. 314–328Google Scholar - 18.M. Babbar, A. Lakshmikantha, D.E. Goldberg, A Modified NSGA-II to solve Noisy Multi-objective Problems, in
*Proceedings of GECCO*(2003)Google Scholar - 19.D.E. Goldberg,
*Genetic Algorithms in Search, Optimization and Machine Learning*(Addison Wesley, 1989)Google Scholar - 20.T. Back, U. Hammel, Evolution strategies applied to perturbed objective functions, in
*Proceedings of 1st IEEE Conference on Evolutionary Computation*, vol. 1, (1994), pp. 40–45Google Scholar - 21.J. Branke, C. Schmidt, H. Schmeck, Efficient fitness estimation in noisy environments, in
*Proceedings of Genetic Evolutionary Computation*(2001), pp. 243–250Google Scholar - 22.C.K. Goh, K.C. Tan, An investigation on noisy environments in evolutionary multiobjective optimization. IEEE Trans. Evol. Comput.
**11**(3), 354–381 (2007)CrossRefGoogle Scholar - 23.D. Buche, P. Stoll, R. Dornberger, P. Koumoutsakos, Multi-objective evolutionary algorithm for the optimization of noisy combustion processes. IEEE Trans. Syst. Man Cybern. Part-C: Appl. Rev.
**32**(4), 460–473 (2002)CrossRefGoogle Scholar - 24.P. Boonma, J. Suzuki, A confidence-based dominance operator in evolutionary algorithms for noisy multiobjective optimization problems, in
*International Conference on Tools with Artificial Intelligence*(2009), pp. 387–394Google Scholar - 25.L. Siwik, S. Natanek, Elitist evolutionary multi-agent system in solving noisy multi-objective optimization problems, in
*IEEE Congress on Evolutionary Computation*, (2008), pp. 3319–3326Google Scholar - 26.C.R. Kube, H. Zhang, The use of perceptual cues in multi-robot box pushing. IEEE Int. Conf. Robot. Autom.
**3**, 2085–2090 (1996)CrossRefGoogle Scholar - 27.A. Verma, B. Jung, G.S. Sukatme, Robot box-pushing with environment-embedded sensors,
*in Proceedings of 2001 IEEE international Symposium on Computational Intelligence in Robotics and Automation*(2001)Google Scholar - 28.J. Chakraborty, A. Konar, A. Nagar, S. Das, Rotation and translation selective Pareto optimal solution to the box-pushing problem by mobile robots using NSGA-II, in
*IEEE CEC*(2009)Google Scholar - 29.T. Robic, B. Philipic, in
*DEMO: Differential Evolution For Multiobjective Optimization*, ed. by C.A. Coello Coello, A.H. Aguirre, E. Zitzler.*Evolutionary*Multi-Criterion Optimization, Third International Conference, EMO 2005. Springer Lecture Notes in Computer Science, vol. 3410 (Guanajuato, Mexico, 2005), pp. 520–533Google Scholar - 30.P. Legendre, D. Borcard, Statistical comparison of univariate tests of homogeneity of variances. J. Stat. Comput. Simul. (2000)Google Scholar
- 31.C.A. Coello Coello, M. Lechuga, MOPSO: a proposal for multiple objective particle swarm optimization. Proc. IEEE Cong. Evol. Comput.
**2**, 1051–1056 (2002)Google Scholar - 32.S. Das, A. Abraham, U.K. Chakraborty, A. Konar, Differential evolution using a neighborhood-based mutation operator. IEEE Trans. Evol. Comput.
**13**(3), 526–553 (2009)CrossRefGoogle Scholar - 33.Standard Error, in Wikipedia, the free EncyclopediaGoogle Scholar
- 34.E.J. Hughes,
*Multi-Objective Probabilistic Selection Evolutionary Algorithm (MOPSEA).*Technical Report No. DAPS/EJH/56/2000, Department of Aerospace, POwer & Sensors, Cranfield University (2000)Google Scholar - 35.E. Zitzler, K. Deb, L. Thiele, Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput.
**8**(2), 173–195 (2000)CrossRefGoogle Scholar - 36.K. Deb, L. Thiele, M. Laumanns, E. Zitzler, Scalable Multi-objective Optimization Test Problems, in
*Proceedings of Congress of Evolutionary Computation*-CEC’02, vol. 1 (Piscataway, New Jersey, 2002), pp. 825–830Google Scholar - 37.Q. Zhang, A. Zhou, S. Zhao, P.N. Suganthan, W. Liu, S. Tiwari, Multi-objective Optimization Test Instances for the CEC 2009 Special Session and Competition. Working Report, CES-887, School of Computer Science and Electrical Engineering, University of Essex, 2008Google Scholar
- 38.S. Huband, P. Hingston, L. Barone, L. While, A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans. Evol. Comput.
**10**(5), 477–506 (2006)CrossRefGoogle Scholar - 39.G.E.P. Box, M.E. Muller, A note on the generation of random deviates. Ann. Math. Stat.
**29**, 610–611 (1958)CrossRefGoogle Scholar - 40.K. Deb, A.P.S. Agarwal, T. Meyarivan, A fast and elitist multi-objective genetic algorithm: NSGA II. IEEE Trans. Evol. Comput.
**2**, 162–197 (1998)CrossRefGoogle Scholar - 41.J.E. Fieldsend,
*Multi-objective Particle Swarm Optimization Methods*(2004)Google Scholar - 42.J.D. Knowles, D. Corne, Approximating the non-dominated front using the pareto archived evolution strategy. Evol. Comput.
**8**(2), 149–172 (2000)CrossRefGoogle Scholar - 43.P. Chakraborty, S. Das, G. Roy, A. Abraham, On convergence of the multi-objective particle swarm optimizers. Inf. Sci.
**181**(8), 1411–1425 (2011)MathSciNetCrossRefGoogle Scholar - 44.C.A. Coello Coello, G.T. Pulido, M.S. Lechuga, Handling multiple objectives with particle swarm optimization. IEEE Trans. Evol. Comput.
**8**(3), 256–279 (2004)CrossRefGoogle Scholar - 45.M. Babbar, A. Lakshmikantha, D.E. Goldberg, A Modified NSGA-II to solve noisy multiobjective problems, in
*Genetic and Evolutionary Computation Conference*Late Breaking Papers, Chicago, IL, pp. 21–27 (2003)Google Scholar - 46.S. Markon, V.D. Arnold, T. Baick, T. Beislstein, G.-H. Beyer, Thresholding—a selection operator for noisy ES, in
*Proceedings of Congress on Evolutionary Computation*(CEC-2001), pp. 465–472Google Scholar - 47.D. Fogel, H.-G. Beyer, A note on the empirical evaluation of intermediate recombination. Evol. Comput.
**3**(4), 491–495 (1995)CrossRefGoogle Scholar - 48.H. Sugie, Y. Inagaki, S. Ono, H. Aisu, T. Unemi, Placing objects with multiple mobile robots-manual help with intension inference in
*IEEE international Conference on Robotics and Automation*(1995), pp. 2181–2186Google Scholar - 49.B. Flury,
*A First Course in Multivariate Statistics*, vol. 28 (Springer, New York, 1997)CrossRefGoogle Scholar - 50.U.K. Chakraborty,
*Advances in Differential Evolution*(Springer, Heidelberg, New York, 2008)CrossRefGoogle Scholar - 51.D. Sheskin,
*Handbook of Parametric and Nonparametric Statistical Procedures*, 4th edn. (Chapman and Hall/CRC, 2007)Google Scholar - 52.J. Derrac, S. Garcia, D. Molina, F. Herrera, A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm and Evol. Comput.
**1**, 3–18 (2011)CrossRefGoogle Scholar - 53.S. Picek, M. Golub, D. Jakobovic, Evaluation of crossover operator performance in genetic algorithms with binary representation, in
*Proceedings of the 7th International Conference on Intelligent Computing: Bio-inspired Computing and Applications*(Springer, Berlin, Heidelberg, 2011), pp. 223–230CrossRefGoogle Scholar - 54.S. Garcia, D. Molina, M. Lozano, F. Herrera, A study on the use of non-parametric tests for analyzing the evolutionary algorithms behavior: a case study on the cec2005 special session on real parameter optimization. J. Heuristics
**15**, 617–644 (2009)CrossRefGoogle Scholar - 55.E.F.P. González, G.R. Torres, G.T. Pulido, Motion-Planning for Co-operative Multi-Robot Box-Pushing problem, in
*Advances in Artificial Intelligence*—*IBERAMIA 2008*. Lecture Notes in Computer Science, vol. 5290, (2008), pp 382–391Google Scholar