Abstract
In this paper, we introduce a sequential approximate strong Karush–Kuhn–Tucker (ASKKT) condition for a multiobjective optimization problem with inequality constraints. We show that each local efficient solution satisfies the ASKKT condition, but weakly efficient solutions may not satisfy it. Subsequently, we use a so-called cone-continuity regularity (CCR) condition to guarantee that the limit of an ASKKT sequence converges to an SKKT point. Finally, under the appropriate assumptions, we show that the ASKKT condition is also a sufficient condition of properly efficient points for convex multiobjective optimization problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andreani R, Haeser G, Martínez JM (2011) On sequential optimality conditions for smooth constrained optimization. Optimization 60:627–641
Andreani R, Haeser G, Ramos A, Silva PJ (2017) A second-order sequential optimality condition associated to the convergence of optimization algorithms. IMA J Numer Anal 37:1902–1929
Andreani R, Haeser G, Schuverdt ML, Silva PJ (2012) Two new weak constraint qualifications and applications. SIAM J Optim 22:1109–1135
Andreani R, Martínez JM, Ramos A, Silva PJ (2016) A cone-continuity constraint qualification and algorithmic consequences. SIAM J Optim 26:96–110
Andreani R, Martínez JM, Svaiter BF (2010) A new sequential optimality condition for constrained optimization and algorithmic consequences. SIAM J Optim 20:3533–3554
Bigi G, Pappalardo M (1999) Regularity conditions in vector optimization. J Optim Theory Appl 102:83–96
Birgin EG, Martínez JM (2014) Practical augmented Lagrangian methods for constrained optimization. Fundam Algorithms, vol 10. SIAM, Philadelphia
Chen L, Goldfarb D (2006) Interior-point \(\ell _2\)-penalty methods for nonlinear programming with strong global convergence properties. Math Program 108:1–36
Dutta J, Deb K, Tulshyan R, Arora R (2013) Approximate KKT points and a proximity measure for termination. J Global Optim 56:1463–1499
Ehrgott M (2005) Multicriteria optimization. Springer Science & Business Media, Berlin
Fiacco AV, McCormick GP (1968) Nonlinear programming: sequential unconstrained minimization techniques. Wiley, New York
Geoffrion AM (1968) Proper efficiency and the theory of vector maximization. J Math Anal Appl 22:618–630
Giorgi G, Jiménez B, Novo V (2016) Approximate Karush–Kuhn–Tucker condition in multiobjective optimization. J Optim Theory Appl 171:70–89
Haeser G, Schuverdt ML (2011) On approximate KKT condition and its extension to continuous variational inequalities. J Optim Theory Appl 149:528–539
Maciel MC, Santos SA, Sottosanto GN (2009) Regularity conditions in differentiable vector optimization revisited. J Optim Theory Appl 142:385–398
Maeda T (1994) Constraint qualifications in multiobjective optimization problems: differentiable case. J Optim Theory Appl 80:483–500
Qi L, Wei Z (2000) On the constant positive linear dependence condition and its application to SQP methods. SIAM J Optim 10:963–981
Wendell RE, Lee DN (1977) Efficiency in multiple objective optimization problems. Math Program 12:406–414
Acknowledgements
We thank anonymous referees for helpful comments. The research was supported by the National Natural Science Foundation of China (Grant Numbers: 11571055, 11601437) and the Fundamental Research Funds for the Central Universities (Grant Number: 106112017CDJZRPY0020).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Feng, M., Li, S. An approximate strong KKT condition for multiobjective optimization. TOP 26, 489–509 (2018). https://doi.org/10.1007/s11750-018-0491-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11750-018-0491-6
Keywords
- Multiobjective optimization
- Strong KKT conditions
- Approximate KKT conditions
- Regularity conditions
- Properly efficient solutions