Abstract
In this paper, an approximation algorithm for solving nonconvex multiobjective programming problems (NCMOPs) is presented. We modify Benson’s method using cones instead of hyperplanes. This algorithm uses an inner approximation and an outer approximation to generate (weakly) efficient solutions and (weakly \(\varepsilon \)-) nondominated points of NCMOPs. Some numerical examples are presented to clarify the proposed algorithm.
Similar content being viewed by others
References
Bartle RG, Donald RS (1992) Introduction to real analysis. Wiley, New York
Benson HP (1998a) An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem. J Glob Optim 13:1–24
Benson HP (1998b) Hybrid approach for solving multiple-objective linear programs in outcome space. J Optim Theory Appl 98:17–35
Ehrgott M, Wiecek MM (2005) Mutiobjective programming. In: Figueira J, Greco S, Ehrgott M (eds) Multiple criteria decision analysis. State of the art surveys. Springer, New York, pp 667–722
Ehrgott M, Schöbel A, Shao L (2010) An approximation algorithm for convex multi-objective programming problems. J Glob Optim 50:397–416
Ehrgott M, Löhne A, Shao L (2012) A dual variant of Benson’s outer approximation algorithm. J Glob Optim 52:757–778
Erfani T, Sergei VU (2011) Directed search domain: a method for even generation of the Pareto frontier in multiobjective optimization. Eng Optim 43:467–484
Fukuda EH, Drummond LG, Raupp FM (2015) An external penalty-type method for multicriteria. TOP 24:1–21
Gourion D, Luc DT (2008) Generating the weakly efficient set of nonconvex multiobjective problems. J Glob Optim 41:517–538
Gourion D, Luc DT (2010) Finding efficient solutions by free disposal outer approximation. SIAM J Optim 20:2939–2958
Hamel AH, Löhne A, Rudloff B (2014) A Benson type algorithm for linear vector optimization and applications. J Glob Optim 59:811–836
Khanh P, Tung L (2015) First- and second-order optimality conditions for multiobjective fractional programming. TOP 23:419440
Löhne A (2011) Vector optimization with infimum and supremum. Springer, Berlin
Löhne A, Rudloff B, Ulus F (2014) Primal and dual approximation algorithms for convex vector optimization problems. J Glob Optim 60:713–736
Ruzika S, Wiecek MM (2005) Approximation methods in multiobjective programming. J Optim Theory Appl 126:473–501
Shao L, Ehrgott M (2008a) Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning. Math Methods Oper Res 68:257–276
Shao L, Ehrgott M (2008b) Approximating the nondominated set of an MOLP by approximately solving its dual problem. Math Methods Oper Res 68:469–492
Acknowledgments
The first-named author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Iran and a Grant from IPM [No. 94900423].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nobakhtian, S., Shafiei, N. A Benson type algorithm for nonconvex multiobjective programming problems. TOP 25, 271–287 (2017). https://doi.org/10.1007/s11750-016-0430-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-016-0430-3