MONSS: A multi-objective nonlinear simplex search approach

References

• Audet, Charles, Gilles Savard, and Walid Zghal. 2010. “A Mesh Adaptive Direct Search Algorithm for Multiobjective Optimization.” European Journal of Operational Research 204 (3): 545–556. doi: 10.1016/j.ejor.2009.11.010
   Web of Science ®Google Scholar
• Bowman, V. J.Jr. 1976. “On the Relationship of the Tchebycheff Norm and the Efficient Frontier of Multiple-Criteria Objectives.” In Vol. 13076-86 of Lecture Notes in Economics and Mathematical Systems. Berlin: Springer.
   Google Scholar
• Chelouah, Rachid, and Patrick Siarry. 2003. “Genetic and Nelder–Mead Algorithms Hybridized for a More Accurate Global Optimization of Continuous Multiminima Functions.” European Journal of Operational Research 148 (2): 335–348. doi: 10.1016/S0377-2217(02)00401-0
   Web of Science ®Google Scholar
• Coello Coello, Carlos A., and Gary B. Lamont, eds. 2004. Applications of Multi-Objective Evolutionary Algorithms. Singapore: World Scientific. doi: 10.1142/5712
   Google Scholar
• Coello Coello, Carlos A., Gary B. Lamont, and David A. Van Veldhuizen. 2007. Evolutionary Algorithms for Solving Multi-Objective Problems. 2nd ed. New York: Springer.
   Google Scholar
• Deb, Kalyanmoy. 1999. “Multi-Objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems.” Evolutionary Computation 7 (3): 205–230. doi: 10.1162/evco.1999.7.3.205
   PubMed Web of Science ®Google Scholar
• Deb, Kalyanmoy, Amrit Pratap, Sameer Agarwal, and T. Meyarivan. 2002. “A Fast and Elitist Multiobjective Genetic Algorithm: NSGA–II.” IEEE Transactions on Evolutionary Computation 6 (2): 182–197. doi: 10.1109/4235.996017
   Web of Science ®Google Scholar
• Deb, Kalyanmoy, Lothar Thiele, Marco Laumanns, and Eckart Zitzler. 2005. “Scalable Test Problems for Evolutionary Multiobjective Optimization.” In Evolutionary Multiobjective Optimization. Theoretical Advances and Applications, edited by Ajith Abraham, Lakhmi Jain, and Robert Goldberg, 105–145. London: Springer.
   Google Scholar
• Deniss, Jhon E.Jr, and Daniel J. Woods. 1987. “Optimization on Microcomputers: The Nelder–Mead Simplex Algorithm.” In New Computing Environments: Microcomputers in Large-scale Computing, edited by Arthur Wouk, 116–122. Philadelphia, PA: SIAM.
   Google Scholar
• Fischer, Andreas, and Pradyumn Kumar Shukla. 2008. “A Levenberg–Marquardt Algorithm for Unconstrained Multicriteria Optimization.” Operations Research Letters 36 (5): 643–646. doi: 10.1016/j.orl.2008.02.006
   Web of Science ®Google Scholar
• Fliege, J., L. M. Graña Drummond, and B. F. Svaiter. 2009. “Newton's Method for Multiobjective Optimization.” SIAM Journal on Optimization 20 (2): 602–626. doi: 10.1137/08071692X
   Web of Science ®Google Scholar
• Fonseca, Carlos M., and Peter J. Fleming. 1995. “Multiobjective Genetic Algorithms Made Easy: Selection, Sharing, and Mating Restriction.” In Proceedings of the First International Conference on Genetic Algorithms in Engineering Systems: Innovations and Applications (GALESIA’95), 12–14 September 1995, Sheffield, UK, 42–52. London: IEE.
   Google Scholar
• Hatcher, Allen. 2001. Algebraic Topology. 1st ed. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Hernández-Díaz, Alfredo G., Carlos A. Coello Coello, Fátima Pérez, Rafael Caballero, Julián Molina, and Luis V. Santana-Quintero. 2008. “Seeding the Initial Population of a Multi-Objective Evolutionary Algorithm Using Gradient-Based Information.” In 2008 Congress on Evolutionary Computation (CEC’2008), June 2008, Hong Kong, 1617–1624. Piscataway, NJ: IEEE.
   Google Scholar
• Hooke, Robert, and T. A. Jeeves. 1961. “‘Direct Search’ Solution of Numerical and Statistical Problems.” Journal of the Association for Computing Machinery 8 (2): 212–229. doi: 10.1145/321062.321069
   Google Scholar
• Koduru, Praveen, Sanjoy Das, and Stephen M. Welch. 2007. “Multi-Objective Hybrid PSO Using ε-Fuzzy Dominance.” In 2007 Genetic and Evolutionary Computation Conference (GECCO’2007), edited by Dirk Thierens, Vol. 1, 853–860. London: ACM Press.
   Google Scholar
• Lagarias, J. C., J. A. Reeds, M. H. Wright, and P. E. Wright. 1998. “Convergence Properties of the Nelder–Mead Simplex Method in Low Dimensions.” SIAM Journal on Optimization 9 (1): 112–147. doi: 10.1137/S1052623496303470
   Web of Science ®Google Scholar
• Laumanns, Marco, Günter Rudolph, and Hans-Paul Schwefel. 1998. “A Spatial Predator–Prey Approach to Multi-Objective Optimization: A Preliminary Study.” In Parallel Problem Solving From Nature—PPSN V, edited by A. E. Eiben, M. Schoenauer, and H.-P. Schwefel, 241–249. Amsterdam, The Netherlands: Springer-Verlag.
   Google Scholar
• Levenberg, K. 1944. “A Method for the Solution of Certain Non-Linear Problems in Least Squares.” Quarterly of Applied Mathematics II (2): 164–168.
   Google Scholar
• Li, Hui, and Qingfu Zhang. 2009. “Multiobjective Optimization Problems with Complicated Pareto Sets, MOEA/D and NSGA-II.” IEEE Transactions on Evolutionary Computation 12 (2): 284–302. doi: 10.1109/TEVC.2008.925798
   Web of Science ®Google Scholar
• Li, Yang, Aimin Zhou, and Guixu Zhang. 2014. “An MOEA/D with Multiple Differential Evolution Mutation Operators.” In 2014 IEEE Congress on Evolutionary Computation (CEC’2014), 6–11 July 2014, Beijing, PR China, 397–404. Piscataway, NJ: IEEE Press.
   Google Scholar
• Lis, Joanna, and A. E. Eiben. 1997. “A Multi-Sexual Genetic Algorithm for Multiobjective Optimization.” In Proceedings of the 1996 International Conference on Evolutionary Computation (ICEC’97), 13–16 April 1997, Indianapolis, IN, 59–64. Piscataway, NJ: IEEE. http://dx.doi.org/10.1109/ICEC.1997.592269.
   Google Scholar
• MacQueen, J. B. 1967. “Some Methods for Classification and Analysis of MultiVariate Observations.” In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, Vol. 1, 281–297. Berkeley, CA: University of California Press. http://projecteuclid.org/euclid.bsmsp/1200512992.
   Google Scholar
• Marquardt, Donald W. 1963. “An Algorithm for Least-Squares Estimation of Nonlinear Parameters.” SIAM Journal on Applied Mathematics 11 (2): 431–441. doi: 10.1137/0111030
   Web of Science ®Google Scholar
• McKinnon, K. I. M. 1998. “Convergence of the Nelder–Mead Simplex Method to a Nonstationary Point.” SIAM Journal on Optimization 9 (1): 148–158. doi: 10.1137/S1052623496303482
   Web of Science ®Google Scholar
• Miettinen, Kaisa. 1999. Nonlinear Multiobjective Optimization. Boston, MA: Kluwer Academic.
   Google Scholar
• Moraglio, Alberto, and Sara Silva. 2011. “Geometric Nelder–Mead Algorithm on the Space of Genetic Programs.” In Proceedings of the 13th Annual Conference on Genetic and Evolutionary Computation (GECCO’11), 12–16 July, Dublin, Ireland, 1307–1314. New York, NY: ACM Press.
   Google Scholar
• Moraglio, A., J. Togelius, and S. Silva. 2013. “Geometric Differential Evolution for Combinatorial and Programs Spaces.” Evolutionary Computation 21 (4): 591–624. doi: 10.1162/EVCO_a_00099
   PubMed Web of Science ®Google Scholar
• Murata, Tadahiko, and Hisao Ishibuchi. 1995. “MOGA: Multi-Objective Genetic Algorithms.” In Proceedings of the 2nd IEEE International Conference on Evolutionary Computing, 29 November–1 December 1995, Perth, Western Australia, 289–294. Piscataway, NJ: IEEE.
   Google Scholar
• Neddermeijer, H. G., Gerrit van Oortmarssen, N. Piersma, Rommert Dekker, and Dik Habbema. 2000. Adaptive Extensions of the Nelder and Mead Simplex Method for Optimization of Stochastic Simulation Models. Econometric Institute Research Papers EI 2000-22/A. Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
   Google Scholar
• Nelder, J. A., and R. Mead. 1965. “A Simplex Method for Function Minimization.” The Computer Journal 7 (4): 308–313. doi: 10.1093/comjnl/7.4.308
   Web of Science ®Google Scholar
• Newton, Isaac. 1669. De Analysi per Aequationes Numero Terminorum Infinitas [On Analysis by Equations with an Infinite Number of Terms].
   Google Scholar
• Parkinson, J. M., and D. Hutchinson. 1972. “An Investigation into the Efficiency of Variants on the Simplex Method.” In Numerical Methods for Non-linear Optimization, edited by F. A. Lootsma, 115–135. London: Academic Press.
   Google Scholar
• Powell, Michael J. D. 1964. “An Efficient Method for Finding the Minimum of a Function of Several Variables Without Calculating Derivatives.” The Computer Journal 7 (2): 155–162. doi: 10.1093/comjnl/7.2.155
   Web of Science ®Google Scholar
• Price, C. J., I. D. Coope, and D. Byatt. 2002. “A Convergent Variant of the Nelder–Mead algorithm.” Journal of Optimization Theory and Applications 113 (1): 5–19. doi: 10.1023/A:1014849028575
   Web of Science ®Google Scholar
• Rahman, M. K. 2006. “An Intelligent Moving Object Optimization Algorithm for Design Problems with Mixed Variables, Mixed Constraints and Multiple Objectives.” Structural and Multidisciplinary Optimization 32 (1): 40–58. doi: 10.1007/s00158-005-0552-y
   Web of Science ®Google Scholar
• Schütze, Oliver, Sanaz Mostaghim, Michael Dellnitz, and Jürgen Teich. 2003. “Covering Pareto Sets by Multilevel Evolutionary Subdivision Techniques.” In Proceedings of the 2nd International Conference on Evolutionary Multi-Criterion Optimization (EMO’03), Faro, Portugal, 118–132. Berlin: Springer-Verlag.
   Google Scholar
• Spendley, W., G. R. Hext, and F. R. Himsworth. 1962. “Sequential Application of Simplex Designs in Optimization and Evolutionary Operation.” Technometrics 4 (4): 441–461. doi: 10.1080/00401706.1962.10490033
   Web of Science ®Google Scholar
• Torczon, Virginia Joanne. 1989. “Multi-Directional Search: A Direct Search Algorithm for Parallel Machines.” PhD thesis, Rice University, Houston, TX.
   Google Scholar
• Trabia, Mohamed B., and Xiao Bin Lu. 2001. “A Fuzzy Adaptive Simplex Search Optimization Algorithm.” Journal of Mechanical Design 123: 216–225. doi: 10.1115/1.1347991
   Web of Science ®Google Scholar
• Valenzuela-Rendón, Manuel, and Eduardo Uresti-Charre. 1997. “A Non-Generational Genetic Algorithm for Multiobjective Optimization.” In Proceedings of the Seventh International Conference on Genetic Algorithms (ICEC’97), East Lansing, MI, 19–23 July 1997, edited by Thomas Bäck, 658–665. San Mateo, CA: Morgan Kaufmann.
   Google Scholar
• Viennet, Rémy, Christian Fontiex, and Ivan Marc. 1996. “Multicriteria Optimization Using a Genetic Algorithm for Determining a Pareto Set.” International Journal of Systems Science 27 (2): 255–260. doi: 10.1080/00207729608929211
   Web of Science ®Google Scholar
• Wang, Zhenkun, Qingfu Zhang, Maoguo Gong, and Aimin Zhou. 2014. “A Replacement Strategy for Balancing Convergence and Diversity in MOEA/D.” In 2014 IEEE Congress on Evolutionary Computation (CEC’2014), 6–11 July 2014, Beijing, PR China, 2132–2139. Piscataway, NJ: IEEE Press.
   Google Scholar
• Wilcoxon, Frank. 1945. “Individual Comparisons by Ranking Methods.” Biometrics Bulletin 1 (6): 80–83. doi: 10.2307/3001968
   Google Scholar
• Xiong, Qiang, and Arthur Jutan. 2003. “Continuous Optimization Using a Dynamic Simplex Method.” Chemical Engineering Science 58 (16): 3817–3828. doi: 10.1016/S0009-2509(03)00236-7
   Web of Science ®Google Scholar
• Zangwill, W. I. 1967. “Minimizing a Function Without Calculating Derivatives.” The Computer Journal 10 (3): 293–296. doi: 10.1093/comjnl/10.3.293
   Web of Science ®Google Scholar
• Zapotecas Martínez, Saúl, and Carlos A. Coello Coello. 2008. “A Proposal to Hybridize Multi-Objective Evolutionary Algorithms with Non-Gradient Mathematical Programming Techniques.” In Parallel Problem Solving from Nature (PPSN X), Vol. 5199 of Lecture Notes in Computer Science, edited by Günter Rudolph, Thomas Jansen, Simon Lucas, Carlo Poloni, and Nicola Beume, 837–846. Dortmund, Germany: Springer.
   Google Scholar
• Zapotecas Martínez, Saúl, and Carlos A. Coello Coello. 2010. “A Memetic Algorithm with Non Gradient-Based Local Search Assisted by a Meta-Model.” In Proceedings of the 11th International Conference on Parallel Problem Solving from Nature (PPSN XI), 11–15 September 2010, Kraków, Poland, Part 1, Vol. 6238 of Lecture Notes in Computer Science, edited by Robert Schaefer, Carlos Cotta, Joanna Kolodziej, and Rudolph Günter, 576–585. Berlin: Springer. http://dx.doi.org/10.1007/978-3-642-15844-5_58.
   Google Scholar
• Zhang, Qingfu, and Hui Li. 2007. “MOEA/D: A Multiobjective Evolutionary Algorithm Based on Decomposition.” IEEE Transactions on Evolutionary Computation 11 (6): 712–731. doi: 10.1109/TEVC.2007.892759
   Web of Science ®Google Scholar
• Zhang, Qingfu, Wudong Liu, and Hui Li. 2009. “The Performance of a New Version of MOEA/D on CEC09 Unconstrained MOP Test Instances.” In 2009 IEEE Congress on Evolutionary Computation (CEC’2009), 18–21 May 2009, Trondheim, Norway, 203–208. Piscataway, NJ: IEEE Press.
   Google Scholar
• Zhang, Q., A. Zhou, S. Zhao, P. N. Suganthan, W. Liu, and S. Tiwari. 2008. Multiobjective Optimization Test Instances for the CEC 2009 Special Session and Competition. Tech. Rep. CES-487, University of Essex and Nanyang Technological University.
   Google Scholar
• Zitzler, Eckart, Kalyanmoy Deb, and Lothar Thiele. 2000. “Comparison of Multiobjective Evolutionary Algorithms: Empirical Results.” Evolutionary Computation 8 (2): 173–195. doi: 10.1162/106365600568202
   PubMed Web of Science ®Google Scholar
• Zitzler, Eckart, Marco Laumanns, and Lothar Thiele. 2001. “SPEA2: Improving the Strength Pareto Evolutionary Algorithm.” In Evolutionary Methods for Design, Optimization and Control with Applications to Industrial Problems (EUROGEN 2001), 19–21 September 2001, Athens, Greece, edited by K. Giannakoglou, D. Tsahalis, J. Periaux, P. Papailou, and T. Fogarty, 95–100. Barcelona: International Center for Numerical Methods in Engineering (CIMNE).
   Google Scholar
• Zitzler, Eckart, and Lothar Thiele. 1998. “Multiobjective Optimization Using Evolutionary Algorithms—A Comparative Study.” In Parallel Problem Solving from Nature V, 27–30 September 1998, edited by Agoston E. Eiben, Thomas Bäck, Marc Schoenauer, and Hans-Paul Schwefel, 292–301. Amsterdam, The Netherlands: Springer-Verlag.
   Google Scholar