Extended well-posedness of quasiconvex vector optimization problems. (English) Zbl 1169.90020

Crespi, G. P.; Papalia, M.; Rocca, M.

J. Optim. Theory Appl. 141, No. 2, 285-297 (2009).

Authors’ 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.

Reviewer: Klaus Schittkowski (Bayreuth)

MSC:

90C29	Multi-objective and goal programming
90C25	Convex programming

Keywords:

vector optimization; multicriteria optimization; quasiconvex problems; well-posedness

References:

[1]	Dontchev, A.L., Zolezzi, T.: Well-Posed Optimization Problems. Lecture Notes in Mathematics, vol. 1543. Springer, Berlin (1993) · Zbl 0797.49001
[2]	Lucchetti, R.: Convexity and Well-posed Problems. Springer, New York (2006) · Zbl 1106.49001
[3]	Hadamard, J.: Sur les problè mes aux dé rivees partielles et leur signification physique. Bull. Univ. Princet. 13, 49–52 (1902)
[4]	Tykhonov, A.N.: On the stability of the functional optimization problem. USSR J. Comput. Math. Math. Phys. 6(4), 631–634 (1966)
[5]	Zolezzi, T.: Extended well-posedness of optimization problems. J. Optim. Theory Appl. 91(1), 257–266 (1996) · Zbl 0873.90094
[6]	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) · Zbl 0848.49017
[7]	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) · Zbl 1119.49025
[8]	Miglierina, E., Molho, E.: Well-posedness and convexity in vector optimization. Math. Methods Oper. Res. 58, 375–385 (2003) · Zbl 1083.90036
[9]	Miglierina, E., Molho, E., Rocca, M.: Well-posedness and scalarization in vector optimization. J. Optim. Theory Appl. 126(2), 391–409 (2005) · Zbl 1129.90346
[10]	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)
[11]	Huang, X.X.: Extended well-posedness properties of vector optimization problems. J. Optim. Theory Appl. 106, 165–182 (2000) · Zbl 1028.90067
[12]	Huang, X.X.: Extended and strongly extended well-posedness of set-valued optimization problems. Math. Methods Oper. Res. 53(1), 101–116 (2001) · Zbl 1018.49019
[13]	Lucchetti, R., Miglierina, E.: Stability for convex vector optimization problems. Optimization 53(5–6), 517–528 (2004) · Zbl 1153.90536
[14]	Luc, D.T.: Theory of Vector Optimization. Springer, Berlin (1989) · Zbl 0688.90051
[15]	Tanino, T.: Stability and sensitivity analysis in convex vector optimization. SIAM J. Control Optim. 26, 521–536 (1988) · Zbl 0654.49011
[16]	Hiriart-Urruty, J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Methods Oper. Res. 4, 79–97 (1979) · Zbl 0409.90086
[17]	Zaffaroni, A.: Degrees of efficiency and degrees of minimality. SIAM J. Control Optim. 42, 1071–1086 (2003) · Zbl 1046.90084
[18]	Levitin, E.S., Polyak, B.T.: Convergence of minimizing sequences in conditional extremum problems. Sov. Math. Dokl. 7, 764–767 (1966) · Zbl 0161.07002