Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints
RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 2, pp. 543-559.

We establish Fritz John necessary conditions for local weak efficient solutions of vector equilibrium problems with constraints in terms of contingent derivatives. Under suitable constraint qualifications, Karush–Kuhn–Tucker necessary conditions for those solutions are investigated.

DOI : 10.1051/ro/2017042
Classification : 90C46, 90C29, 49J52
Mots-clés : Vector equilibrium problems, Local weak efficient solutions, Constraint qualifications, Fritz John and Karush–Kuhn–Tucker efficiency conditions
Luu, Do Van 1 ; Su, Tran Van 1

1 
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     author = {Luu, Do Van and Su, Tran Van},
     title = {Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     pages = {543--559},
     publisher = {EDP-Sciences},
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Luu, Do Van; Su, Tran Van. Contingent derivatives and necessary efficiency conditions for vector equilibrium problems with constraints. RAIRO - Operations Research - Recherche Opérationnelle, Tome 52 (2018) no. 2, pp. 543-559. doi : 10.1051/ro/2017042. http://www.numdam.org/articles/10.1051/ro/2017042/

[1] J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions, in: Advances in Mathematics Supplementary Studies. Academic Press, New York (1981) 160–232 | MR | Zbl

[2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Birkhauser, Boston (1990) | MR | Zbl

[3] P. Daniele, Lagrange multipliers and infinite-dimensional equilibrium problems. J. Glob. Optim. 40 (2008) 65–70 | DOI | MR | Zbl

[4] I.V. Girsanov, Lectures on Mathematical Theory of Extremum Problems. Springer–Verlag, Berlin, Heidenberg (1972) | DOI | MR | Zbl

[5] F. Giannessi, G. Mastroeni and L. Pellegrini, On the theory of vector optimization and variational inequalities, image space analysis and separation, in: Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, edited by F. Giannessi. Kluwer, Dordrecht (2000) 153–215 | DOI | MR | Zbl

[6] X.H. Gong, Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342 (2008) 1455–1466 | DOI | MR | Zbl

[7] X.H. Gong, Scalarization and optimality conditions for vector equilibrium problems. Nonlinear Anal. 73 (2010) 3598–3612 | DOI | MR | Zbl

[8] B. Jiménez and V. Novo, First order optimality conditions in vector optimization involving stable functions. Optimization 57 (2008) 449–471 | DOI | MR | Zbl

[9] B. Jiménez, V. Novo and M. Sama, Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives. J. Math. Anal. Appl. 352 (2009) 788–798 | DOI | MR | Zbl

[10] J. Jahn and R. Rauh, Contingent epiderivatives and set-valued optimization. Math. Meth. Oper. Res. 46 (1997) 193–211 | DOI | MR | Zbl

[11] D.T. Luc, Contingent derivatives of set-valued maps and applications to vector optimization. Math. Program. 50 (1991) 99–111 | DOI | MR | Zbl

[12] D.V. Luu, Convexificators and necessary conditions for efficiency. Optimization 63 (2014) 321–335 | DOI | MR | Zbl

[13] D.V. Luu, Necessary and sufficient conditions for efficiency via convexificators. J. Optim. Theory Appl. 160 (2014) 510–526 | DOI | MR | Zbl

[14] D.V. Luu and D.D. Hang, On optimality conditions for vector variational inequalities. J. Math. Anal. Appl. 412 (2014) 792–804 | DOI | MR | Zbl

[15] D.V. Luu and D.D. Hang, Efficient solutions and optimality conditions for vector equilibrium problems. Math. Methods Oper. Res. 79 (2014) 163–177 | DOI | MR | Zbl

[16] B.C. Ma and X.H. Gong, Optimality conditions for vector equilibrium problems in normed spaces. Optimization 60 (2011) 1441–1455 | DOI | MR | Zbl

[17] P. Michel and J.P. Penot, A generalized derivative for calm and stable functions. Diff. Integ. Equ. 5 (1992) 433–454 | MR | Zbl

[18] J. Morgan and M. Romaniello, Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities. J. Optimiz. Theory Appl. 130 (2006) 309–316 | DOI | MR | Zbl

[19] F. Raciti, Equilibrium conditions and vector variational inequalities: a complex relation. J. Glob. Optimiz. 40 (2008) 353–360 | DOI | MR | Zbl

[20] R.T. Rockafellar, Convex Analysis. Princeton University Press, Princeton (1970) | DOI | MR | Zbl

[21] L. Rodriguez-Marin and M. Sama, About contingent epiderivatives. J. Math. Anal. Appl. 327 (2007) 745–762 | DOI | MR | Zbl

[22] L. Rodriguez-Marin and M. Sama, Variational characterizations of the contingent epiderivatives. J. Math. Anal. Appl. 335 (2007) 1374–1382 | DOI | MR | Zbl

[23] T.V. Su, Optimality conditions for vector equilibrium problems in terms of contingent epiderivatives. Numer. Funct. Anal. Optimiz. 37 (2016) 640–665 | DOI | MR | Zbl

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