Abstract
The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems.
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Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993)
Lucchetti, R.: Convexity and Well-posed Problems. Springer, New York (2006)
Hadamard, J.: Sur les problè mes aux dé rivees partielles et leur signification physique. Bull. Univ. Princet. 13, 49–52 (1902)
Tykhonov, A.N.: On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6(4), 631–634 (1966)
Zolezzi, T.: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91(1), 257–266 (1996)
Loridan, P.: Well-posedness in vector optimization. In: Lucchetti, R., Revalski, J. (eds.) Recent Developments in Well-posed Variational Problems. Mathematics and its Applications, vol. 331, pp. 171–192. Kluwer Academic, Dordrecht (1995)
Crespi, G.P., Guerraggio, A., Rocca, M.: Well posedness in vector optimization problems and vector variational inequalities. J. Optim. Theory Appl. 132(1), 213–226 (2007)
Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58, 375–385 (2003)
Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126(2), 391–409 (2005)
Papalia, M., Rocca, M.: Strong well-posedness and scalarization of vector optimization problems. In: Nonlinear Analysis with Applications in Economics, Energy and Transportation, pp. 209–222. Bergamo University Press—Collana Scienze Matematiche, Statistiche e Informatiche, Bergamo (2007)
Huang, X.X.: Extended well-posedness properties of vector optimization problems. J. Optim. Theory Appl. 106, 165–182 (2000)
Huang, X.X.: Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53(1), 101–116 (2001)
Lucchetti, R., Miglierina, E.: Stability for convex vector optimization problems. Optimization 53(5–6), 517–528 (2004)
Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989)
Tanino, T.: Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim. 26, 521–536 (1988)
Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Methods Oper. Res. 4, 79–97 (1979)
Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003)
Levitin, E.S., Polyak, B.T.: Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Dokl. 7, 764–767 (1966)
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Communicated by F. Giannessi.
Research partially supported by the Cariplo Foundation, Grant 2006.1601/11.0556, Cattaneo University, Castellanza, Italy.
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Crespi, G.P., Papalia, M. & Rocca, M. Extended Well-Posedness of Quasiconvex Vector Optimization Problems. J Optim Theory Appl 141, 285–297 (2009). https://doi.org/10.1007/s10957-008-9494-z
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DOI: https://doi.org/10.1007/s10957-008-9494-z