Abstract
By means of some new results on generalized systems, vector quasi-equilibrium problems with a variable ordering relation are investigated from the image perspective. Lagrangian-type optimality conditions and gap functions are obtained under mild generalized convexity assumptions on the given problem. Applications to the analysis of error bounds for the solution set of a vector quasi-equilibrium problem are also provided. These results are refinements of several authors’ works in recent years and also extend some corresponding results in the literature.
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References
Jahn, J.: Vector Optimization. Theory, Applications, and Extensions. Springer, Heidelberg (2011)
Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)
Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht (2000)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 1–23 (1993)
Ansari, Q.H., Chan, W.K., Yang, X.Q.: The system of vector quasi-equilibrium problems with applications. J. Glob. Optim. 29, 45–57 (2004)
Ansari, Q.H., Yao, J.C.: On vector quasi-equilibrium problems. In: Daniele, P.,Giannessi, F., Maugeri, A. (eds.) Equilibrium Problems and Variational Models, pp. 1–18. Kluwer Academic Publishers, Dordrecht (2003)
Chen, G.Y., Huang, X.X., Yang, X.Q.: Vector Optimization. Set-Valued and Variational Analysis. Springer, Berlin (2005)
Mastroeni, G.: On the image space analysis for vector quasi-equilibrium problems with a variable ordering relation. J. Glob. Optim. 53, 203–214 (2012)
Guu, S.M., Li, J.: Vector quasi-equilibrium problems: separation, saddle points and error bounds for the solution set. J. Glob. Optim. 58, 751–767 (2014)
Li, S.J., Teo, K.L., Yang, X.Q., Wu, S.Y.: Gap functions and existence of solutions to generalized vector quasi-equilibrium problems. J. Glob. Optim. 34, 427–440 (2006)
Mastroeni, G., Panicucci, B., Passacantando, M., Yao, J.C.: A separation approach to vector quasi-equilibrium problems: saddle point and gap function. Taiwan. J. Math. 13, 657–673 (2009)
Mastroeni, G.: Gap functions for equilibrium problems. J. Glob. Optim. 27, 411–426 (2003)
Giannessi, F.: Constrained Optimization and Image Space Analysis. Springer, New York (2005)
Li, J., Mastroeni, G.: Image convexity of generalized systems and applications. J. Optim. Theory Appl. 169, 91–115 (2016)
Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part I: quasi-relative interiors and duality theory. Math. Program. Ser. A 57, 15–48 (1992)
Limber, M.A., Goodrich, R.K.: Quasi interiors, Lagrange multipliers, and \(L^p\) spectral estimation with lattice bounds. J. Optim. Theory Appl. 78, 143–161 (1993)
Bot, R.I., Csetnek, E.R., Wanka, G.: Regularity conditions via quasirelative interior in convex programming. SIAM J. Optim. 19, 217–233 (2008)
Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. Ser. A 122, 301–347 (2010)
Flores-Bazán, F., Mastroeni, G.: Strong duality in cone constrained nonconvex optimization. SIAM J. Optim. 23, 153–169 (2013)
Borwein, J.M., Goebel, R.: Notions of relative interior in Banach spaces. J. Math. Sci. 115, 2542–2553 (2003)
Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, vol. I. Springer, Berlin (1993)
Zhou, Z.A., Yang, X.M.: Optimality conditions of generalized subconvexlike set-valued optimization problems. J. Optim. Theory Appl. 150, 327–340 (2011)
Giannessi, F.: Theorems of the alternative and optimality conditions. J. Optim. Theory Appl. 60, 331–365 (1984)
Mastroeni, G., Rapcsák, T.: On convex generalized systems. J. Optim. Theory Appl. 104, 605–627 (2000)
Bot, R.I., Csetnek, E.R., Moldovan, A.: Revisiting some duality theorems via the quasirelative interior in convex optimization. J. Optim. Theory Appl. 139, 67–84 (2008)
Bot, R.I., Csetnek, E.R.: Regularity conditions via generalized interiority notions in convex optimization: new achievements and their relation to some classical statements. Optimization 61, 35–65 (2012)
Giannessi, F., Mastroeni, G.: Separation of sets and Wolfe duality. J. Glob. Optim. 42, 401–412 (2008)
Holmes, R.B.: Geometric Functional Analysis and Its Applications. Springer, New York (1975)
Li, J., Huang, N.J.: Image space analysis for vector variational inequalities with matrix inequality constraints and applications. J. Optim. Theory Appl. 145, 459–477 (2010)
Li, J., Huang, N.J.: Image space analysis for variational inequalities with cone constraints and applications to traffic equilibria. Sci. China Math. 55, 851–868 (2012)
Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific Publishing Co.,Inc, River Edge (2002)
Acknowledgements
The research was supported by the National Natural Science Foundation of China, Project 11371015; the Innovation Team of Department of Education of Sichuan Province, Project 16TD0019; and the Meritocracy Research Funds of China West Normal University, Project 17YC379.
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Li, J., Mastroeni, G. Refinements on Gap Functions and Optimality Conditions for Vector Quasi-Equilibrium Problems via Image Space Analysis. J Optim Theory Appl 177, 696–716 (2018). https://doi.org/10.1007/s10957-017-1182-4
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DOI: https://doi.org/10.1007/s10957-017-1182-4