Multiobjective Bilevel Programming: Concepts and Perspectives of Development
Bilevel programs model hierarchical non-cooperative decision processes with two decision makers, the leader and the follower, who control different sets of variables and have their own objective functions with interdependent constraints. Bilevel programs are very difficult to solve and even the linear case is NP-hard. In this chapter, a novel view on the main concepts in multiobjective and semivectorial bilevel problems is offered, including new types of solutions that are relevant for decision support. Optimistic and pessimistic leader’s perspectives are explored; the extreme optimistic/deceiving and pessimistic/rewarding solutions in semivectorial problems and the optimistic Pareto fronts in multiobjective problems are defined and illustrated. Traditional and emerging application fields are reviewed. Potential difficulties and pitfalls associated with computing solutions to bilevel models with multiple objectives are outlined, shaping possible research avenues.
KeywordsMultiobjective optimization Bilevel programming Semivectorial bilevel Optimistic versus pessimistic approaches Optimistic Deceiving Pessimistic Rewarding solutions
This work was supported by projects UID/MULTI/00308/2019, ESGRIDS (POCI-01-0145-FEDER-016434) and MAnAGER (POCI-01-0145-FEDER-028040).
- Alves, M. J., & Antunes, C. H. (2016). An illustration of different concepts of solutions in semivectorial bilevel programming. In 2016 IEEE Symposium Series on Computational Intelligence (SSCI) (pp. 1–7). https://doi.org/10.1109/ssci.2016.7850219.
- Alves, M. J., & Antunes, C. H. (2018a). A differential evolution algorithm to semivectorial bilevel problems, machine learning, optimization, and big data. MOD 2017. In G. Nicosia et al. (Eds.), Lecture notes in computer science. Cham: Springer. https://doi.org/10.1007/978-3-319-72926-8_15.
- Alves, M. J., & Antunes, C. H. (2018b). A semivectorial bilevel programming approach to optimize electricity dynamic time-of-use retail pricing. Computers and Operations Research, 92. https://doi.org/10.1016/j.cor.2017.12.014.
- Alves, M. J., Antunes, C. H., & Carrasqueira, P. (2015). A PSO approach to semivectorial bilevel programming: pessimistic, optimistic and deceiving solutions. In Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2015) (pp. 599–606). https://doi.org/10.1145/2739480.2754644.
- Bonnel, H. (2006). Optimality conditions for the semivectorial bilevel optimization problem. Pacific Journal of Optimization, 2(3), 447–468.Google Scholar
- Bostian, M., Whittaker, G., Sinha, A., et al. (2015a). Incorporating data envelopment analysis solution methods into bilevel multi-objective optimization. In 2015 IEEE Congress on Evolutionary Computation (CEC) (pp. 1667–1674).Google Scholar
- Calvete, H. I., & Galé, C. (2010a). A multiobjective bilevel program for production-distribution planning in a supply chain. In Multiple criteria decision making for sustainable energy and transportation systems. Springer, pp. 155–165.Google Scholar
- Carrasqueira, P., Alves, M. J., & Antunes, C. H. (2015). A Bi-level multiobjective PSO algorithm. In A. Gaspar-Cunha, C. H. Antunes & C. Coello Coello (Eds.), Evolutionary Multi-Criterion Optimization (EMO 2015), Lecture notes in computer science 9018 (pp. 263–276). Springer International Publishing.Google Scholar
- Deb, K., & Sinha, A. (2009). Solving bilevel multi-objective optimization problems using evolutionary algorithms. In Proceedings of EMO 2009, LNCS 5467 (pp. 110–124). Springer.Google Scholar
- Deb, K., & Sinha, A. (2010). An efficient and accurate solution methodology for bilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm. Evolutionary computation, 18(3), 403–449.Google Scholar
- Dempe, S. (2002). Foundations of bilevel programming. Springer US.Google Scholar
- Dempe, S. (2009). Bilevel programming: Implicit function approach. In C. A. Floudas & P. M. Pardalos (Eds.), Encyclopedia of optimization (pp. 260–266). Boston, MA: Springer US. https://doi.org/10.1007/978-0-387-74759-0_44.
- Eichfelder, G. (2010). Multiobjective bilevel optimization. Mathematical Programming, 123(2) 419–449.Google Scholar
- Ghotbi, E. (2016). Multi-objective optimization of mechanism design using a bi-level game theoretic formulation. In Concurrent engineering (Vol. 24, No. 3, pp. 266–274). London, England: SAGE Publications Sage UK.Google Scholar
- Gupta, A., & Ong, Y. (2015). An evolutionary algorithm with adaptive scalarization for multiobjective bilevel programs. In 2015 IEEE Congress on Evolutionary Computation (CEC) (pp. 1636–1642). Sendai.Google Scholar
- Halter, W., & Mostaghim, S. (2006). Bilevel optimization of multi-component chemical systems using particle swarm optimization. In 2006 IEEE Congress on Evolutionary Computation (CEC) (pp. 1240–1247).Google Scholar
- Hammad, A. W. A., Rey, D., & Akbarnezhad, A. (2018). A Bi-level mixed integer programming model to solve the multi-servicing facility location problem, minimising negative impacts due to an existing semi-obnoxious facility. In Data and decision sciences in action (pp. 381–395). Springer.Google Scholar
- Lv, Y., & Wan, Z. (2014). A solution method for the optimistic linear semivectorial bilevel optimization problem. Journal of Inequalities and Applications, 2014(1), 164. https://doi.org/10.1186/1029-242x-2014-164.
- Ruuska, S., & Miettinen, K. (2012). Constructing evolutionary algorithms for bilevel multiobjective optimization. In 2012 IEEE Congress on Evolutionary Computation (CEC) (pp. 1–7).Google Scholar
- Sinha, A., et al. (2013). Multi-objective stackelberg game between a regulating authority and a mining company: A case study in environmental economics. In 2013 IEEE Congress on Evolutionary Computation (CEC) (pp. 478–485).Google Scholar
- Sinha, A., Malo, P., & Deb, K. (2015). Transportation policy formulation as a multi-objective bilevel optimization problem. In 2015 IEEE Congress on Evolutionary Computation (CEC) (pp. 1651–1658). https://doi.org/10.1109/cec.2015.7257085.
- Sinha, A., Malo, P., & Deb, K. (2017). Evolutionary bilevel optimization: an introduction and recent advances. In S. Bechikh, R. Datta, & A. Gupta (Eds.), Recent advances in evolutionary multi-objective optimization (pp. 71–103). Cham: Springer.Google Scholar
- Tsoukalas, A., Wiesemann, W., & Rustem, B. (2009). Global optimisation of pessimistic bi-level problems. Fields Institute Communications, 55, 1–29.Google Scholar
- Xu, J., Tu, Y., & Zeng, Z. (2012). A nonlinear multiobjective bilevel model for minimum cost network flow problem in a large-scale construction project. Mathematical Problems in Engineering, 40 (Article ID 463976). https://doi.org/10.1155/2012/463976.