Summary
This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Working in the objective space, necessary and sufficient conditions for Kutateladze’s approximate elements of the image set are given through scalarization in such a way that these points are approximate solutions for a scalar optimization problem. To obtain sufficient conditions we use monotone functions. A new concept is then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved.
This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BFM2003-02194.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990).
Bazaraa, M.S., Sheraly, H.D., Shetty, C.M.: Nonlinear Programming. Theory and Algorithms. John Wiley & Sons, New York (1993).
Deng, S.: On approximate solutions in convex vector optimization. SIAM J. Control Optim., 35, 2128–2136 (1997).
Dutta, J., Vetrivel, V.: On approximate minima in vector optimization. Numer. Funct. Anal. Optim., 22, 845–859 (2001).
Gutiérrez, C., Jiménez, B., Novo, V.: Multiplier rules and saddle-point theorems for Helbig’s approximate solutions in convex Pareto problems. J. Global Optim., 32, (2005).
Gutiérrez, C., Jiménez, B., Novo, V.: A property of efficient and ε-efficient solutions in vector optimization. Appl. Math. Lett., 18, 409–414 (2005).
Helbig, S., Pateva, D.: On several concepts for ε-efficiency. OR Spektrum, 16, 179–186 (1994).
Kutateladze, S.S.: Convex ε-programming. Soviet Math. Dokl., 20, 391–393 (1979).
Liu, J.C.: ε-properly efficient solutions to nondifferentiable multiobjective programming problems. Appl. Math. Lett., 12, 109–113 (1999).
Loridan, P.: ε-solutions in vector minimization problems. J. Optim. Theory Appl., 43, 265–276 (1984).
Li, Z., Wang, S.: ε-approximate solutions in multiobjective optimization. Optimization, 44, 161–174 (1998).
Tammer, Chr.: Stability results for approximately efficient solutions. OR Spektrum, 16, 47–52 (1994).
White, D.J.: Epsilon efficiency. J. Optim. Theory Appl., 49, 319–337 (1986).
Wierzbicki, A.P.: On the completeness and constructiveness of parametric characterizations to vector optimization problems. OR Spektrum, 8, 73–87 (1986).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Gutiérrez, C., Jiménez, B., Novo, V. (2006). Conditions and parametric representations of approximate minimal elements of a set through scalarization. In: Di Pillo, G., Roma, M. (eds) Large-Scale Nonlinear Optimization. Nonconvex Optimization and Its Applications, vol 83. Springer, Boston, MA. https://doi.org/10.1007/0-387-30065-1_11
Download citation
DOI: https://doi.org/10.1007/0-387-30065-1_11
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-30063-4
Online ISBN: 978-0-387-30065-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)