Numerical Algorithms

, Volume 81, Issue 3, pp 915–946 | Cite as

Bilevel optimization with a multiobjective problem in the lower level

  • Roberto Andreani
  • Viviana A. Ramirez
  • Sandra A. SantosEmail author
  • Leonardo D. Secchin
Original Paper


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.


Bilevel optimization Inexact restoration Multiobjective optimization KKT reformulation Numerical experiments 

Mathematics Subject Classification (2010)

90C29 65K05 49M37 


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


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Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of CampinasCampinasBrazil
  2. 2.Academic Regional Center of BarilocheNational University of ComahueSan Carlos de BarilocheArgentina
  3. 3.Department of Applied MathematicsFederal University of Espírito SantoSão MateusBrazil

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