Hostname: page-component-89b8bd64d-5bvrz Total loading time: 0 Render date: 2026-05-07T07:11:55.915Z Has data issue: false hasContentIssue false
  • English
  • Français

Coercivity properties and well-posednessinvector optimization*

Published online by Cambridge University Press:  15 December 2003

Sien Deng
Sien Deng*
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, USA; deng@math.niu.edu.
Get access

Abstract

This paper studies the issue of well-posednessfor vector optimization. It is shown thatcoercivity implies well-posedness without any convexity assumptionson 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 obtaina number of positive results, including a criterion for well-posedness in terms of that of associated scalar problems.In particularwe show that a well-known relative interiority conditioncan be used as a sufficient condition for well-posedness in convexvector optimization.

Keywords

Vector optimizationweakly efficient solutionwell posednesslevel-coercivityerror boundsrelative interior.

Information

Type
Research Article
Information
RAIRO - Operations Research , Volume 37 , Issue 3 , July 2003 , pp. 195 - 208
Copyright
© EDP Sciences, 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Auslender, A., How to deal with the unbounded in optimization: Theory and algorithms. Math. Program. B 79 (1997) 3-18.
Auslender, A., Existence of optimal solutions and duality results under weak conditions. Math. Program. 88 (2000) 45-59. CrossRef
Auslender, A., Cominetti, R. and Crouzeix, J.-P., Convex functions with unbounded level sets and applications to duality theory. SIAM J. Optim. 3 (1993) 669-695. CrossRef
Borwein, J.M. and Lewis, A.S., Partially finite convex programming, Part I: Quasi relative interiors and duality theory. Math. Program. B 57 (1992) 15-48. CrossRef
B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-linear Parametric Optimization. Birhauser-Verlag (1983).
Deng, S., Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization. J. Optim. Theory Appl. 96 (1998) 123-131. CrossRef
Deng, S., On approximate solutions in convex vector optimization. SIAM J. Control Optim. 35 (1997) 2128-2136. CrossRef
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).
Dentcheva, D. and Helbig, S., On variational principles, level sets, well-posedness, and ε-solutions in vector optimization. J. Optim. Theory Appl. 89 (1996) 325-349. CrossRef
L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems. Springer-Verlag, Lecture Notes in Math. 1543 (1993).
F. Flores-Bazan and F. Flores-Bazan, Vector equilibrium problems under recession analysis. preprint, 2001.
Huang, X.X. and Yang, X.Q., 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. CrossRef
Huang, X.X., Pointwise well-posedness of perturbed vector optimization problems in a vector-valued variational principle. J. Optim. Theory Appl. 108 (2001) 671-686. CrossRef
Ioffe, A.D., Lucchetti, R.E. and Revalski, J.P., A variational principle for problems with functional constraints. SIAM J. Optim. 12 (2001) 461-478. CrossRef
Luo, Z.-Q. and Zhang, S.Z., On extensions of Frank-Wolfe theorem. J. Comput. Optim. Appl. 13 (1999) 87-110. CrossRef
D.T. Luc, Theory of Vector Optimization. Springer-Verlag (1989).
R. Lucchetti, Well-posedness, towards vector optimization. Springer-Verlag, Lecture Notes Economy and Math. Syst. 294 (1986).
R.T. Rockafellar, Convex Analysis. Princeton University Press (1970).
R.T. Rockafellar, Conjugate Duality and Optimization. SIAM (1974).
R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag (1998).
Y. Sawaragi, H. Nakayama and T. Tanino, Theory of Multi-objective Optimization. Academic Press (1985).
Zolezzi, T., Well-posedness and optimization under perturbations. Ann. Oper. Res. 101 (2001) 351-361. CrossRef