Abstract
On the basis of the notion of approximating family of cones and a generalized type of Gerstewitz’s/Tammer’s nonlinear scalarization functional, we establish variants of the Ekeland variational principle (for short, EVP) involving set perturbations for a type of approximate proper solutions in the sense of Henig of a vector equilibrium problem. Initially, these results are obtained for both an unconstrained and a constrained vector equilibrium problem, where the objective function takes values in a real locally convex Hausdorff topological linear space. After that, we consider special cases when the objective function takes values in a normed space and in a finite-dimensional vector space. For the finite-dimensional objective space with a polyhedral ordering cone, we give the explicit representation of variants of EVP depending on matrices, and in such a way, some selected applications for multiobjective optimization problems and vector variational inequality problems are also derived.
Similar content being viewed by others
Notes
f is said to be (q, H)-lower semicontinuous from above ((q, H)-lsca) at \(x\in X\) if, for each \(r \in \mathbb {R}\) and \(x_m\rightarrow x\), from \(f( x_1 )+ rq \le _H 0\) and \(f(x_{m+1})+ t_m q \le _H f(x_m)\), \(t_m\ge 0\), for all \(m \in \mathbb {N}\), it follows that \(f(x) + rq \le _H 0\). It is firstly introduced by the authors in [34].
References
Al-Homidan, S., Ansari, Q.H., Kassay, G.: Vectorial form of Ekeland variational principle with applications to vector equilibrium problems. Optimization 69, 415–436 (2020)
Bao, T.Q., Cobzaş, S., Soubeyran, A.: Variational principles, completeness and the existence of traps in behavioral sciences. Ann. Oper. Res. 269, 53–79 (2018)
Borwein, J.M., Zhuang, D.: Super efficiency in vector optimization. Trans. Am. Math. Soc. 338, 105–122 (1993)
Bednarczuk, E.M., Zagrodny, D.: Vector variational principle. Archiv der Mathematik 93, 577–586 (2009)
Bednarczuk, E.M., Przybyla, M.J.: The vector-valued variational principle in Banach spaces ordered by cones with nonempty interiors. SIAM J. Optim. 18, 907–913 (2007)
Benson, H.P.: An improved definition of proper efficiency for vector maximization with respect to cones. J. Math. Anal. Appl. 71, 232–241 (1979)
Bianchi, M., Kassay, G., Pini, R.: Ekeland’s principle for vector equilibrium problems. Nonlinear Anal. 66, 1454–1464 (2007)
Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Am. Math. Soc. 16, 605–611 (1965)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
El Maghri, M.: Pareto-Fenchel \(\varepsilon \)-subdifferential sum rule and \(\varepsilon \)-efficiency. Optim. Lett. 6, 763–781 (2012)
Finet, C., Quarta, L.: Vector-valued perturbed equilibrium problems. J. Math. Anal. Appl. 343, 531–545 (2008)
Flores-Baz\(\acute{\rm a}\)n, F., Hern\(\acute{\rm a}\)ndez, E.: A unified vector optimization problem: complete scalarizations and applications. Optimization. 60, 1399-1419 (2011)
Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)
Gong, X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)
Gong, X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007)
Göpfert, A., Riahi, H., Tammer, C., Zălinescu, C.: Variational Methods in Partially Ordered Spaces. Springer, Berlin (2003)
Göpfert, A., Tammer, C., Zălinescu, C.: On the vectorial Ekeland’s variational principle and minimal points in product spaces. Nonlinear Anal. 39, 909–922 (2000)
Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and \(\varepsilon \)-subdifferentials in vector optimization: basic properties and limit behaviour. Nonlinear Anal. 79, 52–67 (2013)
Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Proper approximate solutions and \(\varepsilon \)-subdifferentials in vector optimization: Moreau-Rockafellar type theorems. J. Convex. Anal. 21, 857–886 (2014)
Gutiérrez, C., Huerga, L., Jiménez, B., Novo, V.: Henig approximate proper efficiency and optimization problems with difference of vector mappings. J. Convex. Anal. 23, 661–690 (2016)
Gutiérrez, C., Huerga, L., Novo, V.: Scalarization and saddle points of approximate proper solutions in nearly subconvexlike vector optimization problems. J. Math. Anal. Appl. 389, 1046–1058 (2012)
Gutiérrez, C., Huerga, Novo, V.: Nonlinear scalarization in multiobjective optimization with a polyhedral ordering cone. Int. Trans. Oper. Res. 45, 763–779 (2017)
Gutiérrez, C., Huerga, L., Novo, V., Sama, M.: Limit behavior of approximate proper solutions in vector optimization. SIAM J. Optim. 29, 2677–2696 (2020)
Gutiérrez, C., Jiménez, B., Novo, V.: On approximate efficiency in multiobjective programming. Math. Method Oper. Res. 64, 165–185 (2006)
Gutiérrez, C., Jiménez, B., Novo, V.: A set-valued Ekeland’s variational principle in vector optimization. SIAM J. Control. Optim. 47, 883–903 (2008)
Gutiérrez, C., Jiménez, B., Novo, V.: A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM J. Optim. 17, 688–710 (2006)
Gutierrez, C., Novo, V., Ródenas-Pedregosa, J.L., Tanaka, T.: Nonconvex separation functional in linear spaces with applications to vector equilibria. SIAM J. Optim. 26, 2677–2695 (2016)
Gutiérrez, C., Kassay, G., Novo, V., Ródenas-Pedregosa, J.L.: Ekeland variational principles in vector equilibrium problems. SIAM J. Optim. 27, 2405–2425 (2017)
Hai, L.P., Huerga, L., Khanh, P.Q., Novo, V.: Variants of the Ekeland variational principle for approximate proper solutions of vector equilibrium problems. J. Glob. Optim. 74, 361–382 (2019)
Helbig, S.: On a New Concept for \(\varepsilon \)-Efficiency. “ Talk at Optimization Days 1992”, Montreal, (1992)
Henig, M.I.: Proper efficiency with respect to cones. J. Optim. Theory Appl. 36, 387–407 (1982)
Jahn, J.: Vector Optimization: Theory. Applications and Extensions. Springer, Berlin (2011)
Kaliszewski, I.: Quantitative Pareto Analysis by Cone Separation Technique. Kluwer Academic Publishers, Boston (1994)
Khanh, P.Q., Quy, D.N.: A generalized distance and Ekeland’s variational principle for vector functions. Nonlinear Anal. 73, 2245–2259 (2010)
Khanh, P.Q., Quy, D.N.: On generalized Ekeland’s variational principle and equivalent formulations for set-valued mappings. J. Glob. Optim. 49, 381–396 (2011)
Khanh, P.Q., Quy, D.N.: Versions of Ekeland’s variational principle involving set perturbations. J. Glob. Optim. 57, 951–968 (2013)
Kutateladze, K.K.: Convex \(\varepsilon \)-programming. Soviet Math. Dokl. 20, 391–393 (1979)
Liu, C.G., Ng, K.F.: Ekeland’s variational principle for set-valued functions. SIAM J. Optim. 21, 41–56 (2011)
Li, Z., Wang, S.: \(\varepsilon \)-approximate solutions in multiobjective optimization. Optimization 44, 161–174 (1998)
Mordukhovich, B.S.: Variational Analysis and Applications, vol. 8. Springer, Cham (2018)
Németh, A.B.: A nonconvex vector minimization problem. Nonlinear Anal. 10, 669–678 (1986)
Qiu, J.H.: An equilibrium version of vectorial Ekeland variational principle and its applications to equilibrium problems. Nonlinear Anal. Real World Appl. 27, 26–42 (2016)
Qiu, J.H.: Set-valued quasi-metrics and a general Ekeland’s variational principle in vector optimization. SIAM J Control Optim. 51, 1350–1371 (2013)
Qiu, J.H.: A pre-order principle and set-valued Ekeland variational principle. J. Math. Anal. Appl. 419, 904–937 (2014)
Qiu, J.H.: Generalized Gerstewitz’s functions and vector variational principle for \(\varepsilon \)-efficient solutions in the sense of Németh. Acta Mathematica Sinica, English Series 35, 297–320 (2019)
Qiu, J.H., He, F.: Ekeland variational principles for set-valued functions with set perturbations. Optimization 69, 925–960 (2020)
Sterna-Karwat, A.: Approximating families of cones and proper efficiency in vector optimization. Optimization 20, 809–817 (1989)
Ródenas-Pedregosa, J. L.: Caracterización de Soluciones de Problemas de Equilibrio Vectoriales. Doctoral Thesis, UNED, Madrid, (2018)
Tammer, C.: A generalization of Ekeland’s variational principle. Optimization 25, 129–141 (1992)
Tanaka, T.: A new approach to approximation of solutions in vector optimization problems. In: Proceedings of APORS, World Scientific, Singapore (1995)
Vályi, I.: Approximate saddle-point theorems in vector optimization. J. Optim. Theory Appl. 55, 435–448 (1987)
White, D.J.: Epsilon efficiency. J. Optim. Theory Appl. 49, 319–337 (1986)
Acknowledgements
The author is grateful to the anonymous referees for their valuable remarks and suggestions. This work was supported by the Domestic Master/PhD Scholarship Programme of Vingroup Innovation Foundation [Grant Number VINIF.2019.TS.19] and The National Foundation for Science and Technology development (NAFOSTED) under grant no. 101.01-2020.23.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Hai, L.P. Ekeland variational principles involving set perturbations in vector equilibrium problems. J Glob Optim 79, 733–756 (2021). https://doi.org/10.1007/s10898-020-00945-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-020-00945-5
Keywords
- Ekeland variational principle
- Set perturbations
- Approximate proper efficiency
- Approximating family of cones
- Multiobjective programming
- Variational inequalities