# Bilevel optimization with a multiobjective problem in the lower level

- 64 Downloads

## Abstract

Bilevel problems model instances with a hierarchical structure. Aiming at an efficient solution of a constrained multiobjective problem according with some pre-defined criterion, we reformulate this semivectorial bilevel optimization problem as a classic bilevel one. This reformulation intents to encompass all the objectives, so that the properly efficient solution set is recovered by means of a convenient weighted-sum scalarization approach. Inexact restoration strategies potentially take advantage of the structure of the problem under consideration, being employed as an alternative to the Karush-Kuhn-Tucker reformulation of the bilevel problem. Genuine multiobjective problems possess inequality constraints in their modeling, and these constraints generate theoretical and practical difficulties to our lower level problem. We handle these difficulties by means of a perturbation strategy, providing the convergence analysis, together with enlightening examples and illustrative numerical tests.

## Keywords

Bilevel optimization Inexact restoration Multiobjective optimization KKT reformulation Numerical experiments## Mathematics Subject Classification (2010)

90C29 65K05 49M37## Preview

Unable to display preview. Download preview PDF.

## Notes

### Acknowledgements

The authors are thankful to an anonimous reviewer, whose remarks and suggestions have provide improvements upon the original version of the manuscript.

## References

- 1.Andreani, R., Castro, S.L.C., Chela, J.L., Friedlander, A., Santos, S.A.: An inexact-restoration method for nonlinear bilevel programming problems. Comput. Optim. Appl.
**43**(3), 307–328 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl.
**71**(1), 232–241 (1979)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Benson, H.P.: Optimization over the efficient set. J. Math. Anal. Appl.
**98**(2), 562–580 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Bonnel, H., Morgan, J.: Semivectorial bilevel optimization problem: penalty approach. J. Optim. Theory Appl.
**131**(3), 365–382 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Bonnel, H., Morgan, J.: Optimality conditions for semivectorial bilevel convex optimal control problems, Proceedings in Mathematics & Statistics, vol. 50, pp. 5–78. Springer, Heidelberg (2013)zbMATHGoogle Scholar
- 6.Borwein, J.: Proper efficient points for maximizations with respect to cones. SIAM J. Control. Optim.
**15**(1), 57–63 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Bskens, C., Wassel, D.: The ESA NLP Solver WORHP, chap. 8. Springer. Springer Optimization and Its Applications
**73**, 85–110 (2012)CrossRefGoogle Scholar - 8.Bueno, L.F., Haeser, G., Martínez, J.M.: An inexact restoration approach to optimization problems with multiobjective constraints under weighted-sum scalarization. Optim. Lett.
**10**(6), 1315–1325 (2016)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Deb, K.: Multi-objective genetic algorithms: problem difficulties and construction of test problems. Evol. Comput.
**7**(3), 205–230 (1999)MathSciNetCrossRefGoogle Scholar - 10.Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)zbMATHGoogle Scholar
- 11.El-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the newton interior-point method for nonlinear programming. J. Optim. Theory Appl.
**89**(3), 507–541 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Fülöp, J.: On the equivalence between a linear bilevel programming problem and linear optimization over the efficient set. Technical report WP93-1, Laboratory of Operations Research and Decision Systems, Computer and Automation Institute, Hungarian Academy of Sciences (1993)Google Scholar
- 13.Fletcher, R., Leyffer, S.: Numerical experience with solving MPECs as NLPs. Tech. rep., University of Dundee Report NA pp. 210 (2002)Google Scholar
- 14.Fletcher, R., Leyffer, S.: Solving mathematical programs with complementarity constraints as nonlinear programs. Optimization Methods and Software
**19**(1), 15–40 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim.
**17**(1), 259–286 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl.
**22**(3), 618–630 (1968)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Guo, X.L., Li, S.J.: Optimality conditions for vector optimization problems with difference of convex maps. J. Optim. Theory Appl.
**162**(3), 821–844 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 18.HSL. A collection of Fortran codes for large scale scientific computation. Available at http://www.hsl.rl.ac.uk/
- 19.Huband, S., Barone, L., While, L., Hingston, P.: A scalable multi-objective test problem toolkit, pp. 280–295. Springer, Berlin (2005)zbMATHGoogle Scholar
- 20.Kuhn, H.W., Tucker, A.W.: Nonlinear programming. In: Neyman, J. (ed.) Proceedings of the second Berkeley symposium on mathematical statistics and probability, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
- 21.Martínez, J.M.: Inexact-restoration method with Lagrangian tangent decrease and new merit function for nonlinear programming. J. Optim. Theory Appl.
**3**(1), 39–58 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Martínez, J.M., Pilotta, E.A.: Inexact-restoration algorithm for constrained optimization. J. Optim. Theory Appl.
**104**(1), 135–163 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Martínez, J.M., Svaiter, B.F.: A practical optimality condition without constraint qualifications for nonlinear programming. J. Optim. Theory Appl.
**118**(1), 117–133 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Miettinen, K.M.: Nonlinear Multiobjective Optimization. Kluwer Academic Publishers, Boston (1999)zbMATHGoogle Scholar
- 25.Pilotta, E.A., Torres, G.A.: An inexact restoration package for bilevel programming problems. Appl. Math.
**15**(10A), 1252–1259 (2012)CrossRefGoogle Scholar - 26.Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim.
**11**(4), 918–936 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Srinivas, N., Deb, K.: Muiltiobjective optimization using nondominated sorting in genetic algorithms. Evol. Comput.
**2**(3), 221–248 (1994)CrossRefGoogle Scholar - 28.Tanaka, M., Watanabe, H., Furukawa, Y., Tanino, T.: GA-based decision support system for multicriteria optimization. In: IEEE International Conference on Systems, Man and Cybernetics. Intelligent Systems for the 21St Century, vol. 2, pp. 1556–1561 (1995)Google Scholar
- 29.Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput.
**8**(2), 173–195 (2000)CrossRefGoogle Scholar