Abstract
Real world engineering optimization problems often involve discrete variables (e.g., categorical variables) characterizing choices such as the type of material to be used or the presence of certain system components. From an analytical perspective, these particular variables determine the definition of the objective and constraint functions, as well as the number and type of parameters that characterize the problem. Furthermore, due to the inherent discrete and potentially non-numerical nature of these variables, the concept of metrics is usually not definable within their domain, thus resulting in an unordered set of possible choices. Most modern optimization algorithms were developed with the purpose of solving design problems essentially characterized by integer and continuous variables and by consequence the introduction of these discrete variables raises a number of new challenges. For instance, in case an order can not be defined within the variables domain, it is unfeasible to use optimization algorithms relying on measures of distances, such as Particle Swarm Optimization. Furthermore, their presence results in non-differentiable objective and constraint functions, thus limiting the use of gradient-based optimization techniques. Finally, as previously mentioned, the search space of the problem and the definition of the objective and constraint functions vary dynamically during the optimization process as a function of the discrete variables values.
This paper presents a comprehensive survey of the scientific work on the optimization of mixed-variable problems characterized by continuous and discrete variables. The strengths and limitations of the presented methodologies are analyzed and their adequacy for mixed-variable problems with regards to the particular needs of complex system design is discussed, allowing to identify several ways of improvements to be further investigated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abdelkhalik, O.: Autonomous planning of multigravity-assist trajectories with deep space Maneuvers using a differential evolution approach. Int. J. Aerosp. Eng. 2013, 1–11 (2013)
Abdelkhalik, O.: Hidden genes genetic optimization for variable-size design space problems. J. Optim. Theor. Appl. 156(2), 450–468 (2013)
Abramson, M.A.: Mixed variable optimization of a load-bearing thermal insulation system using a filter pattern search algorithm. Optim. Eng. 5(2), 157–177 (2004)
Abramson, M.A., Audet, C., Chrissis, J.W., Walston, J.G.: Mesh adaptive direct search algorithms for mixed variable optimization. Optim. Lett. 3(1), 35–47 (2009)
Abramson, M.A., Audet, C., Dennis Jr., J.E.: Filter pattern search algorithms for mixed variable constrained optimization problems. Pac. J. Optim. 3(3), 477–500 (2007)
Audet, C., Dennis Jr., J.E.: Pattern search algorithms for mixed variable programming. SIAM J. Optim. 11(3), 573–594 (2000)
Audet, C., Dennis Jr., J.E.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Optim. 17(1), 188–217 (2006)
Bartz-Beielstein, T., Zaefferer, M.: Model-based methods for continuous and discrete global optimization. Appl. Soft Comput. 55, 154–167 (2017)
Coello, C.A.C.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput. Meth. Appl. Mech. Eng. 191(11–12), 1245–1287 (2002)
Deb, K., Pratap, A., Agarwal, S.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)
Dorigo, M.: Optimization, learning and natural algorithms. Ph.D. thesis, Politecnico di Milano, Italy (1992)
Emmerich, M., Grötzner, M., Groß, B., Schütz, M.: Mixed-integer evolution strategy for chemical plant optimization with simulators. In: Evolutionary Design and Manufacture, pp. 55–67. Springer, London (2000)
Coelho, R.F.: Metamodels for mixed variables based on moving least squares. Optim. Eng. 15(2), 311–329 (2014)
Frank, C., Marlier, R., Pinon-Fischer, O.J., Mavris, D.N.: An evolutionary multi-architecture multi-objective optimization algorithm for design space exploration. In: 57th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Reston, Virginia, January 2016. American Institute of Aeronautics and Astronautics
Goldberg, D.E.: Genetic Algorithms in Search Optimization & Machine Learning. Addison-Wesley Longman Publishing Co., Inc., Boston (1989)
Herrera, M., Guglielmetti, A., Xiao, M., Coelho, R.F.: Metamodel-assisted optimization based on multiple Kernel regression for mixed variables. Struct. Multi. Optim. 49(6), 979–991 (2014)
Hooke, R., Jeeves, T.A.: Direct search solution of numerical, statistical problems. J. ACM 8(2), 212–229 (1961)
Isebor, O.J.: Derivative-free optimization for generalized oil field development. Ph.D. thesis, Stanford University, USA (2010)
Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of ICNN 1995 - International Conference on Neural Networks, pp. 1942–1948 (1995)
Kirkpatrick, S., Gelatt, D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)
Kokkolaras, M., Audet, C., Dennis Jr., J.E.: Mixed variable optimization of the number and composition of heat intercepts in a thermal insulation system. Optim. Eng. 2(1), 5–29 (2001)
Land, A.H., Doig, A.G.: An automatic method for solving discrete programming problems. In: 50 Years of Integer Programming 1958–2008, pp. 105–132. Springer, Heidelberg (2010)
Li, R., Emmerich, M.T.M., Eggermont, J., Bovenkamp, E.G.P., Bäck, T., Dijkstra, J., Reiber, J.H.C.: Metamodel-assisted mixed integer evolution strategies and their application to intravascular ultrasound image analysis. In: 2008 IEEE Congress on Evolutionary Computation, CEC 2008, Hong Kong, June 2008. IEEE, pp. 2764–2771
Liao, C.-J., Tseng, C.-T., Luarn, P.: A discrete version of particle swarm optimization for flowshop scheduling problems. Comput. Oper. Res. 34(10), 3099–3111 (2007)
Liao, T., Socha, K., de Oca, M.A.M., Stutzle, T., Dorigo, M.: Ant colony optimization for mixed-variable optimization problems. IEEE Trans. Evol. Comput. 18(4), 503–518 (2014)
Lucidi, S., Piccialli, V., Sciandrone, M.: An algorithm model for mixed variable programming. SIAM J. Optim. 15(4), 1057–1084 (2005)
Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7(4), 308–313 (1964)
Nyew, H.M., Abdelkhalik, O., Onder, N.: Structured-chromosome evolutionary algorithms for variable-size autonomous interplanetary trajectory planning optimization. J. Aerosp. Inf. Syst. 12(3), 314–328 (2015)
Roy, S., Moore, K., Hwang, J.T., Gray, J.S., Crossley, W.A., Martins, J.: A mixed integer efficient global optimization algorithm for the simultaneous aircraft allocation-mission-design problem. In: 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Reston, Virginia, January 2017. American Institute of Aeronautics and Astronautics
Stelmack, M., Nakashima, N., Batill, S.: Genetic algorithms for mixed discrete/continuous optimization in multidisciplinary design. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Reston, Virigina, September 1998. American Institute of Aeronautics and Astronautics
Storn, R., Price, K.: Differential evolution a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11(4), 341–359 (1997)
Sun, C., Zeng, J., Pan, J.-S.: A modified particle swarm optimization with feasibility-based rules for mixed-variable optimization problems. Int, J. Innov. Comput. Inf. Contr. 7(6), 3081–3096 (2011)
Talbi, E.-G.: Metaheuristics: From Design to Implementation. Wiley, New York (2009)
Venter, G., Sobieszczanski-Sobieski, J.: Multidisciplinary optimization of a transport aircraft wing using particle swarm optimization. Struct. Multi. Optim. 26(1–2), 121–131 (2004)
Wang, J., Yin, Z.: A ranking selection-based particle swarm optimizer for engineering design optimization problems. Struct. Multi. Optim. 37(2), 131–147 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this paper
Cite this paper
Pelamatti, J., Brevault, L., Balesdent, M., Talbi, EG., Guerin, Y. (2018). How to Deal with Mixed-Variable Optimization Problems: An Overview of Algorithms and Formulations. In: Schumacher, A., Vietor, T., Fiebig, S., Bletzinger, KU., Maute, K. (eds) Advances in Structural and Multidisciplinary Optimization. WCSMO 2017. Springer, Cham. https://doi.org/10.1007/978-3-319-67988-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-67988-4_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67987-7
Online ISBN: 978-3-319-67988-4
eBook Packages: EngineeringEngineering (R0)