Abstract
In this article, we view the Pareto and weak Pareto solutions of the multiobjective optimization by using an approximate version of KKT type conditions. In one of our main results Ekeland’s variational principle for vector-valued maps plays a key role. We also focus on an improved version of Geoffrion proper Pareto solutions and it’s approximation and characterize them through saddle point and KKT type conditions.
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References
Andreani R, Haeser G, Martínez JM (2011) On sequential optimality conditions for smooth constrained optimization. Optimization 60(5):627–641
Bazaraa MS, Sherali HD, Shetty CM (2013) Nonlinear programming: theory and algorithms. Wiley, New York
Beltran F, Cuate O, Schutze O (2020) The Pareto tracer for general inequality constrained multi-objective optimization problems. Math Comput Appl 25:80. https://doi.org/10.3390/mca25040080
Bennet G, Peitz S (2021) An efficient descent method for locally Lipschitz multiobjective problem. J Optim Th Appl 188:696–723
Braun M, Seijo S, Echanobe J, Shukla PK, Campo I, Garcia-Sedano J, Schmeck H (2016) A neuro-genetic approach for modeling and optimizing a complex cogeneration process. Appl Soft Comput 48:347–358
Braun M, Shukla PK, Schmeck H (2015) Obtaining optimal Pareto front approximations using scalarized preference information. Proceedings of the 2015 Annual Conference on Genetic and Evolutionary Computation, pp 631–638
Chankong V, Haimes YY (2008) Multiobjective decision making: theory and methodology. Courier Dover Publications, New York
Chuong TD, Kim DS (2016) Approximate solutions of multiobjective optimization problems. Positivity 20(1):187–207
Clarke FH (1983) Optimization and nonsmooth analysis, vol 5. Wiley, New York (Republished by SIAM 1990)
Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, New York
Deb K, Tewari R, Dixit M, Dutta J (2007) Finding trade-off solutions close to KKT points using evolutionary multiobjective optimization. Proceedings of the 2007 IEEE Congress on Evolutionary Computation, pp 2109–2116
Deb K, Abouhawwash M, Dutta J (2015) An optimality theory based proximity measure for evolutionary multiobjective and many objective optimization. Lecture Notes Comput Sci 9019:18–33
Dhara A, Dutta J (2011) Optimality conditions in convex optimization: a finite-dimensional view. CRC Press, Cambridge
Durea M, Dutta J, Tammer C (2011) Stability properties of KKT points in vector optimization. Optimization 60(7):823–838
Dutta J (2012) Strong KKT, second order conditions and non-solid cones in vector optimization. In: Recent Developments in Vector Optimization, pp 127–167. Springer
Dutta J, Deb K, Tulshyan R, Arora R (2013) Approximate KKT points and a proximity measure for termination. J Global Optim 56(4):1463–1499
Dutta J, Vetrivel V (2001) On approximate minima in vector optimization. Numer Funct Anal Optim 22(7–8):845–859
Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, Berlin
Eichfelder G (2008) Adaptive scalarization methods in multiobjective optimization, vol 436. Springer, Berlin
Eichfelder G (2014) Variable ordering structures in vector optimization. Springer, Berlin
Eichfelder G, Warnow L (2021) Proximity measures based on KKT points for constrained multi-objective optimization. J Global Optim 80:63–86
Fukuda EH, Grana Drummond LM (2014) A survey on multiobjective descent methods. Pesquisa Operacional 34(3):585–620
Giorgi G, Jiménez B, Novo V (2016) Approximate Karush–Kuhn–Tucker condition in multiobjective optimization. J Optim Theory Appl 171(1):70–89
Göpfert Alfred, Riahi Hassan, Tammer Christiane, Zalinescu Constantin (2006) Variational methods in partially ordered spaces. Springer, Berlin
Goochi G, Liuzzi G, Luidi S, Sciandrone M (2020) On the convergence of steepest descent methods for multiobjective optimization. Comput Opt Appl. https://doi.org/10.1007/s10589-020-00192-0
Gutiérrez C, Jiménez B, Novo V (2006) On approximate efficiency in multiobjective programming. Math Methods Oper Res 64(1):165–185
Gutiérrez C, Jiménez B, Novo V (2010) Optimality conditions via scalarization for a new \(\varepsilon \)-efficiency concept in vector optimization problems. Eur J Oper Res 201(1):11–22
Hillermeirer C (2001) Nonlinear multiobjective optimization-a generalized homotopy approach. Birkhauser
Jahn J (2004) Vector optimization : theory, applications, and extensions. Springer, Berlin
Khan AA, Tammer C, Zalinescu C (2016) Set-valued optimization. Springer, Berlin
Kesarwani P, Shukla PK, Dutta J, Deb K (2019) Approximations for Pareto and Proper Pareto Solutions and their KKT conditions. Optimization Online, http://www.optimization-online.org/DB _HTML/2018/10/6845.html
Kuhn HW, Tucker AW (1951) Nonlinear Programming. Proceedings of the 2nd Berkeley symposium on Mathematical Statistics
Liu JC (1999) \(\varepsilon \)-properly efficient solutions to nondifferentiable multiobjective programming problems. Appl Math Lett 12(6):109–113
Loridan P (1984) \(\varepsilon \)-solutions in vector minimization problems. J Optim Theory Appl 43(2):265–276
Luc DT (1989) Theory of vector optimization. Springer, Berlin
Martin A, Schutze O (2018) Pareto Tracer: a predictor-corrector method for multiobjective optimization problem. Eng Optim 50(3):516–536
Miettinen K (1999) Nonlinear Programming. Kluwer Academic Publishers, Norwell
Mordukhovich BS, Nam NM (2013) An easy path to convex analysis and applications, vol 6. Morgan & Claypool Publishers, San Rafael
Mordukhovich Boris S (2006) Variational analysis and generalized differentiation I: Basic theory, vol 330. Springer, Berlin
Rockafellar RT (1970) Convex analysis. Princeton University Press, Oxford
Rockafellar RT, Wets RJB (1998) Variational analysis, vol 317. Springer, Berlin
Shukla PK, Dutta J, Deb K, Kesarwani P (2019) On a practical notion of geoffrion proper optimality in multicriteria optimization. Optimization, pp 1–27
Shukla PK, Braun M (2013) Indicator based search in variable orderings: theory and algorithms. Lect Notes Comput Sci 7811:66–80
Tammer C (1992) A generalization of Ekeland’s variational principle. Optimization 25(2–3):129–141
Tanabe H, Fukuda EH, Yamashita N (2019) Proximal gradient methods for multiobjective optimization and application. Comput Optim Appl 72(2):339–361
Valyi I (1985) Approximate solutions of vector optimization problems. Annu Rev Autom Program 12:246–250
Acknowledgements
The authors are very grateful to the anonymous referees for their constructive comments which has greatly improved the presentation of the paper. We are also very thankful to the Associate Editor for bringing to our notice the reference Beltran et al. (2020).
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Kesarwani, P., Shukla, P.K., Dutta, J. et al. Approximations for Pareto and Proper Pareto solutions and their KKT conditions. Math Meth Oper Res 96, 123–148 (2022). https://doi.org/10.1007/s00186-022-00787-9
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DOI: https://doi.org/10.1007/s00186-022-00787-9