Coercivity properties and well-posedness in vector optimization*
RAIRO - Operations Research (2010)
- Volume: 37, Issue: 3, page 195-208
- ISSN: 0399-0559
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topDeng, Sien. "Coercivity properties and well-posedness in vector optimization*." RAIRO - Operations Research 37.3 (2010): 195-208. <http://eudml.org/doc/105289>.
@article{Deng2010,
abstract = {
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.
},
author = {Deng, Sien},
journal = {RAIRO - Operations Research},
keywords = {Vector optimization;
weakly efficient solution;
well posedness; level-coercivity; error bounds;
relative interior.; vector optimization; weakly efficient solution; well-posedness; relative interior},
language = {eng},
month = {3},
number = {3},
pages = {195-208},
publisher = {EDP Sciences},
title = {Coercivity properties and well-posedness in vector optimization*},
url = {http://eudml.org/doc/105289},
volume = {37},
year = {2010},
}
TY - JOUR
AU - Deng, Sien
TI - Coercivity properties and well-posedness in vector optimization*
JO - RAIRO - Operations Research
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 3
SP - 195
EP - 208
AB -
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.
LA - eng
KW - Vector optimization;
weakly efficient solution;
well posedness; level-coercivity; error bounds;
relative interior.; vector optimization; weakly efficient solution; well-posedness; relative interior
UR - http://eudml.org/doc/105289
ER -
References
top- A. Auslender, How to deal with the unbounded in optimization: Theory and algorithms. Math. Program. B79 (1997) 3-18.
- A. Auslender, Existence of optimal solutions and duality results under weak conditions. Math. Program.88 (2000) 45-59.
- 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.
- J.M. Borwein and A.S. Lewis, Partially finite convex programming, Part I: Quasi relative interiors and duality theory. Math. Program. B57 (1992) 15-48.
- B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-linear Parametric Optimization. Birhauser-Verlag (1983).
- S. Deng, Characterizations of the nonemptiness and compactness of solution sets in convex vector optimization. J. Optim. Theory Appl.96 (1998) 123-131.
- S. Deng, On approximate solutions in convex vector optimization. SIAM J. Control Optim.35 (1997) 2128-2136.
- 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).
- 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.
- 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.
- 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.
- 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.
- 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.
- Z.-Q. Luo and S.Z. Zhang, On extensions of Frank-Wolfe theorem. J. Comput. Optim. Appl.13 (1999) 87-110.
- 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).
- T. Zolezzi, Well-posedness and optimization under perturbations. Ann. Oper. Res.101 (2001) 351-361.
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