Geodesic
A Note on Extended Characterization of Generalized Trade-Off Directions in Multiobjective Optimization
Journal of convex analysis, Tome 19 (2012) no. 1, pp. 91-111.

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We consider five basic optimality principles for a general multiobjective optimization problem with convex and nonconvex partial objectives. We introduce a generalization of the concept of trade-off directions defining them via some optimal surface of appropriate cones. In both cases, we link optimality and generalized trade-off directions by deriving geometrical optimality conditions in terms of appropriate cones. We scrutinize similarities and differences between the cases. Combining newly proven facts with some previously known results, we derive four general patterns reflecting structural properties and interconnections of the considered optimality principles. Additionally, we provide extended characterization of optimality for some peculiar cases.
Mots-clés : Multiobjective optimization, optimality principles, generalized trade-offs, tangent cones, normal cones, contingent cones, convex and nonconvex optimization 
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     author = {M. M. M\"akel\"a and Y. Nikulin and J. Mezei},
     title = {A {Note} on {Extended} {Characterization} of {Generalized} {Trade-Off} {Directions} in {Multiobjective} {Optimization}},
     journal = {Journal of convex analysis},
     pages = {91--111},
     publisher = {mathdoc},
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     number = {1},
     year = {2012},
     url = {http://geodesic.mathdoc.fr/item/JCA_2012_19_1_JCA_2012_19_1_a4/}
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M. M. Mäkelä; Y. Nikulin; J. Mezei. A Note on Extended Characterization of Generalized Trade-Off Directions in Multiobjective Optimization. Journal of convex analysis, Tome 19 (2012) no. 1, pp. 91-111. http://geodesic.mathdoc.fr/item/JCA_2012_19_1_JCA_2012_19_1_a4/