×

zbMATH — the first resource for mathematics

Necessary optimality conditions for a bilevel multiobjective programming problem via a \(\Psi\)-reformulation. (English) Zbl 1427.90251
Summary: In this paper, we are concerned with a bilevel multiobjective optimization problem \((P)\). First, using \(\Psi\), a function introduced by N. Gadhi and S. Dempe [J. Optim. Theory Appl. 155, No. 1, 100–114 (2012; Zbl 1267.90130)], we transform \((P)\) into a one level optimization problem \((P^\ast)\). Second, on terms of convexificators, using a scalarization technique, we derive a Karash-Kuhn-Tucker (KKT)-type necessary optimality conditions to the initial problem \((P)\) under a generalized Abadie constraint qualification without the assumption that the lower-level problem satisfies the Mangasarian Fromovitz constraint qualification. Some examples have been introduced to illustrate our results.

MSC:
90C29 Multi-objective and goal programming
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bard, JF., Optimality conditions for the bilevel programming problem, Nav Res Logist Q, 31, 13-26, (1984) · Zbl 0537.90087
[2] Bard, JF., Practical bilevel optimization: algorithms and applications, (1998), Dordrecht: Kluwer Academic, Dordrecht
[3] Dempe, S., Foundations of bilevel programming, (2002), Dordrecht: Kluwer Academic, Dordrecht · Zbl 1038.90097
[4] Dempe, S.; Gadhi, N., Second order optimality conditions for bilevel set optimization problems, J Global Optim, 47, 233-245, (2010) · Zbl 1190.90179
[5] Dempe, S.; Dutta, J., Is bilevel programming a special case of a mathematical program with complementarity constraints?, Math Program, 131, 37-48, (2012) · Zbl 1235.90145
[6] Dempe, S.; Franke, S., On the solution of convex bilevel optimization problems, Comput Optim Appl, 63, 685-703, (2016) · Zbl 1343.90065
[7] Outrata, JV., On necessary optimality conditions for Stackelberg problems, J Optim Theory Appl, 76, 306-320, (1993) · Zbl 0802.49007
[8] Ye, JJ., Constraint qualification and KTT conditions for bilevel programming problems, Oper Res, 31, 211-824, (2006)
[9] Eichfelder, G., Multiobjective bilevel optimization, Math Program, 123, 2, 419-449, (2010) · Zbl 1198.90347
[10] Jun, G.; Yan, T.; Benjamin, L., A multi-objective bilevel location planning problem for stone industrial parks, Comput Oper Res, 56, 8-21, (2015) · Zbl 1348.90390
[11] Shiqi, F.; Ping, G.; Mo, L., Bilevel multiobjective programming applied to water resources allocation, Math Probl Eng, 2013, (2013)
[12] Babahadda, H.; Gadhi, N., Necessary optimality conditions for bilevel optimisation problems using convexificators, J Global Optim, 34, 535-549, (2006) · Zbl 1090.49021
[13] Demyanov, VF; Jeyakumar, V., Huntting for a smaller convex subdifferential, J Optim Theory Appl, 10, 305-326, (1997) · Zbl 0872.90083
[14] Dutta, J.; Chandra, S., Convexificators, generalized convexity and optimality conditions, J Optim Theory Appl, 113, 41-65, (2002) · Zbl 1172.90500
[15] Jeyakumar, V.; Luc, T., Nonsmooth calculus, minimality and monotonicity of convexificators, J Optim Theory Appl, 101, 599-621, (1999) · Zbl 0956.90033
[16] Kohli, B., Optimality conditions for optimistic bilevel programming problem using convexificators, J Optim Theory Appl, 152, 632-651, (2012) · Zbl 1262.90137
[17] Li, XF; Zhang, JZ., Necessary optimality conditions in terms of convexificators in Lipschitz optimization, J Optim Theory Appl, 131, 429-452, (2006) · Zbl 1143.90035
[18] Suneja, SK; Kohli, B., Optimality and duality results for bilevel programming problem using convexificators, J Optim Theory Appl, 150, 1-19, (2011) · Zbl 1229.90207
[19] Gadhi, N.; Dempe, S., Necessary optimality conditions and a new approach to multi-objective bilevel optimization problems, J Optim Theory Appl, 155, 100-114, (2012) · Zbl 1267.90130
[20] Hiriart-Urruty, JB., Tangent cones, generalized gradients and mathematical programming in Banach spaces, Math Oper Res, 4, 79-97, (1979) · Zbl 0409.90086
[21] Ciligot-Travain, M., On Lagrange-Kuhn-Tucker multipliers for pareto optimisation problems, Numer Funct Anal Optim, 15, 689-693, (1994) · Zbl 0831.49021
[22] Benkenza, N.; Gadhi, N.; Lafhim, L., Necessary and sufficient optimality conditions for set-valued optimisation, Georgian Math J, 18, 53-66, (2011) · Zbl 1236.90109
[23] Clarke, FC., Optimization and nonsmooth analysis, (1983), New York: Wiley-Interscience, New York
[24] Penot, M., A generalized derivatives for calm and stable functions, Differ Integr Equ, 5, 433-454, (1992) · Zbl 0787.49007
[25] Mordukhovich, BS; Shao, Y., A nonconvex subdifferential calculus in Banach space, J Convex Anal, 2, 211-227, (1995) · Zbl 0838.49013
[26] Dempe, S.; Gadhi, N., Necessary optimality conditions for bilevel set optimization problems, J Global Optim, 39, 4, 529-542, (2007) · Zbl 1190.90178
[27] Hiriart-Urruty, JB; Lemarechal, C., Fundamentals of convex analysis, (2001), Berlin Heidelberg: Springer-Verlag, Berlin Heidelberg · Zbl 0998.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.