Coercivity properties and well-posedness in vector optimization
RAIRO - Operations Research - Recherche Opérationnelle, Volume 37 (2003) no. 3, p. 195-208

This paper studies the issue of well-posedness for vector optimization. It is shown that coercivity implies well-posedness without any convexity assumptions on problem data. For convex vector optimization problems, solution sets of such problems are non-convex in general, but they are highly structured. By exploring such structures carefully via convex analysis, we are able to obtain a number of positive results, including a criterion for well-posedness in terms of that of associated scalar problems. In particular we show that a well-known relative interiority condition can be used as a sufficient condition for well-posedness in convex vector optimization.

DOI : https://doi.org/10.1051/ro:2003021
Keywords: vector optimization, weakly efficient solution, well posedness, level-coercivity, error bounds, relative interior
@article{RO_2003__37_3_195_0,
     author = {Deng, Sien},
     title = {Coercivity properties and well-posedness in vector optimization},
     journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {3},
     year = {2003},
     pages = {195-208},
     doi = {10.1051/ro:2003021},
     zbl = {1070.90095},
     mrnumber = {2034539},
     language = {en},
     url = {http://www.numdam.org/item/RO_2003__37_3_195_0}
}
Deng, Sien. Coercivity properties and well-posedness in vector optimization. RAIRO - Operations Research - Recherche Opérationnelle, Volume 37 (2003) no. 3, pp. 195-208. doi : 10.1051/ro:2003021. http://www.numdam.org/item/RO_2003__37_3_195_0/

[1] A. Auslender, How to deal with the unbounded in optimization: Theory and algorithms. Math. Program. B 79 (1997) 3-18. | MR 1464758 | Zbl 0887.90131

[2] A. Auslender, Existence of optimal solutions and duality results under weak conditions. Math. Program. 88 (2000) 45-59. | MR 1765892 | Zbl 0980.90063

[3] A. Auslender, R. Cominetti and J.-P. Crouzeix, Convex functions with unbounded level sets and applications to duality theory. SIAM J. Optim. 3 (1993) 669-695. | MR 1244046 | Zbl 0808.90102

[4] J.M. Borwein and A.S. Lewis, Partially finite convex programming, Part I: Quasi relative interiors and duality theory. Math. Program. B 57 (1992) 15-48. | MR 1167406 | Zbl 0778.90049

[5] B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-linear Parametric Optimization. Birhauser-Verlag (1983). | MR 701243 | Zbl 0502.49002

[6] S. Deng, Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization. J. Optim. Theory Appl. 96 (1998) 123-131. | MR 1608034 | Zbl 0897.90163

[7] S. Deng, On approximate solutions in convex vector optimization. SIAM J. Control Optim. 35 (1997) 2128-2136. | MR 1478657 | Zbl 0891.90142

[8] S. Deng, Well-posed problems and error bounds in optimization, in Reformulation: Non-smooth, Piecewise Smooth, Semi-smooth and Smoothing Methods, edited by Fukushima and Qi. Kluwer (1999). | MR 1682724 | Zbl 0948.90130

[9] D. Dentcheva and S. Helbig, On variational principles, level sets, well-posedness, and ϵ-solutions in vector optimization. J. Optim. Theory Appl. 89 (1996) 325-349. | MR 1387272 | Zbl 0853.90101

[10] L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems. Springer-Verlag, Lecture Notes in Math. 1543 (1993). | MR 1239439 | Zbl 0797.49001

[11] F. Flores-Bazan and F. Flores-Bazan, Vector equilibrium problems under recession analysis. preprint, 2001. | MR 1814037

[12] X.X. Huang and X.Q. Yang, Characterizations of nonemptiness and compactness of the set of weakly efficient solutions for convex vector optimization and applications. J. Math. Anal. Appl. 264 (2001) 270-287. | MR 1876733 | Zbl 1018.90048

[13] X.X. Huang, Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle. J. Optim. Theory Appl. 108 (2001) 671-686. | MR 1828678 | Zbl 1017.90098

[14] A.D. Ioffe, R.E. Lucchetti and J.P. Revalski, A variational principle for problems with functional constraints. SIAM J. Optim. 12 (2001) 461-478. | MR 1885571 | Zbl 1023.49021

[15] Z.-Q. Luo and S.Z. Zhang, On extensions of Frank-Wolfe theorem. J. Comput. Optim. Appl. 13 (1999) 87-110. | MR 1704115 | Zbl 1040.90536

[16] D.T. Luc, Theory of Vector Optimization. Springer-Verlag (1989). | MR 1116766

[17] R. Lucchetti, Well-posedness, towards vector optimization. Springer-Verlag, Lecture Notes Economy and Math. Syst. 294 (1986).

[18] R.T. Rockafellar, Convex Analysis. Princeton University Press (1970). | MR 274683 | Zbl 0193.18401

[19] R.T. Rockafellar, Conjugate Duality and Optimization. SIAM (1974). | MR 373611 | Zbl 0296.90036

[20] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag (1998). | MR 1491362 | Zbl 0888.49001

[21] Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multi-objective Optimization. Academic Press (1985). | MR 807529 | Zbl 0566.90053

[22] T. Zolezzi, Well-posedness and optimization under perturbations. Ann. Oper. Res. 101 (2001) 351-361. | MR 1852519 | Zbl 0996.90081