Vector critical points and efficiency in vector optimization with Lipschitz functions
Tóm tắt
In this work, we establish some relations between several notions of vector critical points and efficient, weak efficient and ideal efficient solutions of a vector optimization problem with a locally Lipschitz objective function. These relations are stated under pseudoinvexity hypotheses and via the generalized Jacobian. We provide a characterization of pseudoinvexity (resp. strong pseudoinvexity) through the property that every vector critical point is a weak efficient (resp. efficient) solution. We also obtain some properties of invex functions in connection with linear scalarizations. Several examples illustrating our results are also provided.
Tài liệu tham khảo
Arana-Jiménez, M., Rufián-Lizana, A., Osuna-Gómez, R., Ruiz-Garzón, G.: A characterization of pseudoinvexity in multiobjective programming. Math. Comput. Model. 48, 1719–1723 (2008)
Ben-Israel, A., Mond, B.: What is invexity? J. Aust. Math. Soc. Ser. B 28, 1–9 (1986)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
Guerraggio, A., Luc, D.T.: Optimality conditions for C\(^{1,1}\) vector optimization problems. J. Optim. Theory Appl. 109, 615–629 (2001)
Gutiérrez, C., Jiménez, B., Novo, V., Ruiz-Garzón, G.: Vector variational-like inequalities and multiobjective optimization with Lipschitz functions. Submitted (2014)
Martin, D.H.: The essence of invexity. J. Optim. Theory Appl. 47, 65–76 (1985)
Martínez-Legaz, J.E.: What is invexity with respect to the same \(\eta \). Taiwan. J. Math. 13, 753–755 (2009)
Mishra, S.K., Wang, S.Y.: Vector variational-like inequalities and non-smooth vector optimization problems. Nonlinear Anal. 64, 1939–1945 (2006)
Osuna-Gómez, R., Beato-Moreno, A., Rufián-Lizana, A.: Generalized convexity in multiobjective programming. J. Math. Anal. Appl. 233, 205–220 (1999)
Osuna-Gómez, R., Rufián-Lizana, A., Ruíz-Canales, P.: Invex functions and generalized convexity in multiobjective programming. J. Optim. Theory Appl. 98, 651–661 (1998)
Phuong, T.D., Sach, P.H., Yen, N.D.: Strict lower semicontinuity of the level sets and invexity of a locally Lipschitz function. J. Optim. Theory Appl. 87, 579–594 (1995)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Ruiz-Garzón, G., Osuna-Gómez, R., Rufián-Lizana, A.: Relationships between vector variational-like inequality and optimization problems. Eur. J. Oper. Res. 157, 113–119 (2004)
Santos, L.B., Ruiz-Garzón, G., Rojas-Medar, M.A., Rufián-Lizana, A.: Some relations between variational-like inequality problems and vectorial optimization problems in Banach spaces. Comput. Math. Appl. 55, 1808–1814 (2008)