Abstract
A two-phase numerical process is proposed for gradient-based multi-disciplinary optimization. In a first phase, one or several Pareto-optimal solutions associated with a subset of the cost functions, the primary objective functions, subject to constraints, are determined by some effective multi-objective optimizer. In the second phase, a continuum of Nash equilibria is constructed tangent to the Primary Pareto Front along which secondary cost functions are to be reduced, while best preserving the Pareto-optimality of the primary cost functions. The focus of the article is on estimating the rate at which the secondary cost functions are diminished. The method is illustrated by the numerical treatment of the optimal sizing problem of a sandwich panel w.r.t. structural resistance under bending loads and blast mitigation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.P. Aubin, Mathematical Methods of Game and Economic Theory (Courier Corporation, 2007)
I. Das, J. Dennis, Normal-Boundary Intersection: An Alternate Method for Generating Pareto Optimal Points in Multicriteria Optimization Problems, ICASE Report 96–62, ICASE (1996)
K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Transactions on Evolutionary Computation 6 (2002), pp. 182–197
J.A. Désidéri, MGDA Software Platform for Multi-Objective Differentiable Optimization, http://mgda.inria.fr
J.A. Désidéri, Révision de l’algorithme de descente à gradients multiples (MGDA) par orthogonalisation hiérarchique, Research Report 8710, Inria (2015). https://hal.inria.fr/hal-01139994
J.A. Désidéri, Platform for Prioritized Multi-objective Optimization by metamodel-Assisted Nash Games, Research Report 9290, Inria (2019). https://hal.inria.fr/hal-02285197
J.A. Désidéri, R. Duvigneau, Direct and Adpative Approaches to Multi-objective Optimization, Research Report 9291, Inria (2019). https://hal.inria.fr/hal-02285899
J.A. Désidéri, R. Duvigneau, DA. Habbal, Computational Intelligence in Aerospace Sciences, V. M. Becerra and M. Vassile Eds., Progress in Astronautics and Aeronautics, T. C. Lieuwen Ed.-in-Chief Vol. 244, chap. Multi-Objective Design Optimization Using Nash Games, American Institute for Aeronautics and Astronautics Inc., Reston, Virginia (2014)
J.A. Désidéri, P. Leite, Q. Mercier, Prioritized Multi-objective Optimization of a Sandwich Panel, Research Report 9362, Inria, 2020. https://hal.inria.fr/hal-02931770
M. Emmerich, K. Yang, A. Deutz, H. Wang, C.M. Fonseca, Advances in Stochastic and Deterministic Global Optimization, P. M. Pardalos and A. Zhigljavsky and J. Žilinskas eds., chap. A Multicriteria Generalization of Bayesian Global Optimization, Springer Optimization and Its Applications, Springer, Cham (2016)
J. Fliege, B.F. Svaiter, Steepest Descent Methods for Multicriteria Optimization, Math. Methods Oper. Rese., pp. 479–494 (2000)
M.B. Giles, N.A. Pierce, An Introduction to the Adjoint Approach to Design, Flow, Turbulence and Combustion, pp. 393–415 (2000)
D. Greiner, J. Périaux, J.M. Emperador, B. Galván, G. Winter, Game Theory Based Evolutionary Algorithms: A Review with Nash Applicationss in Structural Engineering Optimization Problems, Arch Comput Methods Eng (Springer), pp. 703–750 (2017)
M. Hartikainen and K. Miettinen, Constructing a pareto front approximation for decision making. Math. Methods Oper. Rese., 209–234 (2011)
M. Hartikainen, K. Miettinen, M.M. Wiecek, PAINT: Pareto front interpolation for nonlinear multiobjective optimization, Computational Optimization and Applications (2012), pp. 845–867
L. Hascoët, Tapenade, version 3, https://team.inria.fr/ecuador/fr/tapenade/
J. Horn, N. Nafpliotis, D.E. Goldberg, A niched Pareto genetic algorithm for multiobjective optimization, in Proceedings of the First IEEE Conference on Evolutionary Computation. IEEE World Congress on Computational Intelligence (1991)
A. Jameson, K. Leoviriyakit, and S. Shankaran, Multi-point Aero-Structural Optimization of Wings Including Planform Variations, in 45th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 8–11. 2007
A. Jameson, J.C. Vassberg, S. Shankaran, Aerodynamic-structural design studies of low-sweep transonic wings. J. Aircraft, 47 (2010)
D.R. Jones, M. Schonlau, and W.J. Welch, Efficient global optimization of expensive black-box functions, Journal of Global Optimization 13 (1998), pp. 455–492
P. Leite, Conception architecturale appliquée aux matériaux sandwichs pour propriétés multifonctionnelles, Ph.D. diss., Université de Grenoble, École Doctorale I-MEP2, 2013. http://www.theses.fr/2013GRENI053
J.R.R.A. Martins, A.B. Lambe, Multidisciplinary design optimization: A survey of architectures. AIAA J., 51 (2013)
A. Messac, A. Isamail-Yahaya, C. Mattson, The normalized normal constraint method for generating the pareto frontier. Struct. Multidisciplinary Optim., 86–98 (2003)
T. Murata, H. Hishibuchi, MOGA: multi-objective genetic algorithms, in Proceedings of 1995 IEEE International Conference on Evolutionary Computation, Perth, Australia, pp. 289–294 (1995)
K. Parsopoulos, M. Vrahatis, Recent approaches to global optimization problems through particle swarm optimization. Natural Comput., 235–306 (2002)
J. Périaux, F. Gonzalez, D.S.C. Lee, Evolutionary optimization and game strategies for advanced multi-disciplinary design - applications to aeronautics and UAV design. Intelli. Syst. Control Autom. Sci. Eng. 75, Springer (2015)
J. Sobieszczanski-Sobieski, T.D. Altus, M. Phillips, R. Sandusky,Bilevel integrated system synthesis for concurrent and distributed processing. AIAA J., 41 (2003)
E. Zitzler and L. Thiele, Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach, IEEE Transactions on Evolutionary Computation 3 (1999), pp. 257–271
Acknowledgements
The author wishes to express his warmest thanks to his colleagues from Inria and the University Côte d’Azur R. Duvigneau, A. Habbal, L. Hascoët, L. Monasse and B. Mourrain for very fruitful scientific discussions on Nash games and polytope exploration in large dimension, as well as software implementation of algorithms.
Special thanks are also due to the Inria Service of Experimentation and Development (SED), T. Kloczko, N. Niclausse and J.Wintz particularly, for the development of the web interface of the software platform http://mgda.inria.fr.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix: Configuration of Convex Hulls
Appendix: Configuration of Convex Hulls
See the Fig. 8.
Various geometrical configurations of the convex hull for two and three gradients vectors—In this affine-space representation, the vectors of the convex hull are given the origin O and are pointing over the green line or triangle
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Désidéri, JA. (2021). Adaptation by Nash Games in Gradient-Based Multi-objective/Multi-disciplinary Optimization. In: Nachaoui, A., Hakim, A., Laghrib, A. (eds) Mathematical Control and Numerical Applications. JANO'13 2021. Springer Proceedings in Mathematics & Statistics, vol 372. Springer, Cham. https://doi.org/10.1007/978-3-030-83442-5_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-83442-5_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-83441-8
Online ISBN: 978-3-030-83442-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)