Abstract
We study the optimization problems \(\min _\mathbf{x }\left\{ f(\mathbf{x }):g_j(\mathbf{x })\le 0, j=1,\ldots ,m\right\} \) where Slater’s condition holds without the convexity of the feasible set and of the functions \(f,~ g_j\). At a feasible point \(\mathbf{x }\) under question the functions \(g_j\) are assumed to satisfy a non-degeneracy assumption, necessary and sufficient KKT optimality conditions are then considered in relation to the convexity of the level sets of f.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Slater’s constraint qualification condition holds if there exists \(\mathbf{x }\in \mathbb {R}^n\) such that \(g_j(\mathbf{x })<0\) for all \(j=1,\ldots ,m.\)
References
Ben-Tal, A., Nemirovski, A.: Lectures on modern convex optimization: analysis, algorithms and engineering applications. SIAM, Philadelphia (2001)
Berkovitz, L.D.: Convexity and optimization in \(\mathbb{R}^n\). Wiley, New York (2002)
Bertsekas, D., Nedi\(\acute{c}\), A., Ozdaglar, E.: Convex analysis and optimization. Athena Scientific, Belmont (2003)
Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press, Cambridge (2004)
Hiriart-Urruty, J.-B.: Optimisation et Analyse Convexe. Presses Universitaires de France, Paris (1998)
Nesterov, Y., Nemirovski, A.: Interior-point polynomial algorithms in convex programming. SIAM, Philadelphia (1994)
Arrow, K.J., Enthoven, A.C.: Quasi-concave programming. Econometrica 29(4), 779–800 (1961)
Arrow, K.J., Hurwicz, L., Uzawa, H.: Constraint qualifications in maximization problems. Nav. Res. Logist. 8(2), 175–191 (1961)
Lasserre, J.B.: On representations of the feasible set in convex optimization. Optim. Lett. (2010). doi:10.1007/s11590-009-0153-6
Lasserre, J.B.: On convex optimization without convex representation. Optim. Lett. (2011). doi:10.1007/s11590-011-0323-1
Giorgi, G.: Optimality conditions under generalized convexity revisited. In: Annals of the University of Bucharest (mathematical series) 4(LXII), pp. 479–490 (2013)
Dutta, J., Lalitha, C.S.: Optimality conditions in convex optimization revisited. Optim. Lett. 7(2), 221–229 (2013)
Khanh, P.Q., Quyen, H.T., Yao, J.-C.: Optimality conditions under relaxed quasi-convexity assumptions using star and adjusted subdifferentials. Eur. J. Oper. Res. 212, 235–241 (2011)
Martínez-Legaz, J.E.: Optimality conditions for pseudo-convex minimization over convex sets defined by tangentially convex constraints. Optim. Lett. (2014). doi:10.1007/s11590-014-0822-y
Polyak, B.T.: Introduction to optimization. Optimization Software Inc., New York (1987)
Acknowledgments
We would like to thank the anonymous referees whose suggestions have vastly improved the paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ho, Q. Necessary and sufficient KKT optimality conditions in non-convex optimization. Optim Lett 11, 41–46 (2017). https://doi.org/10.1007/s11590-016-1054-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-016-1054-0