Stability of Local Efficiency in Multiobjective Optimization
Analyzing the behavior and stability properties of a local optimum in an optimization problem, when small perturbations are added to the objective functions, are important considerations in optimization. The tilt stability of a local minimum in a scalar optimization problem is a well-studied concept in optimization which is a version of the Lipschitzian stability condition for a local minimum. In this paper, we define a new concept of stability pertinent to the study of multiobjective optimization problems. We prove that our new concept of stability is equivalent to tilt stability when scalar optimizations are available. We then use our new notions of stability to establish new necessary and sufficient conditions on when strict locally efficient solutions of a multiobjective optimization problem will have small changes when correspondingly small perturbations are added to the objective functions.
KeywordsMultiobjective programming Variational analysis Tilt stability Weighted sum scalarization
Mathematics Subject Classification90C29 90C31 49K40
The authors are indebted to Prof. Boris Mordukhovich for his serious discussions on the earlier draft of this manuscript to enrich this manuscript. Moreover, the authors are grateful for the editor and the two anonymous referees for their valuable and constructive comments to improve this manuscript. They would also like to extend their thankfulness to Maxie Schmidt and Karim Rezaei for the linguistic editing of the final draft of this paper.
- 4.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. II: Applications Grundlehren Math. Springer, Berlin (2006)Google Scholar
- 5.Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, Grundlehren Math. Wiss., vol. 317. Springer, Berlin (2006)Google Scholar
- 12.Mordukhovich, B.S.: Sensitivity analysis in nonsmooth optimization. In: Field, D.A., Komkov, V. (eds.) Theoretical Aspects of Industrial Design, SIAM Volume in Applied Mathematics, vol. 58, pp. 32–46 (1992)Google Scholar
- 15.Bokrantz, R., Fredriksson, A.: On solutions to robust multiobjective optimization problems that are optimal under convex scalarization. arXiv:1308.4616v2 (2014)
- 23.Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. CMS Book Math, vol. 20. Springer, New York (2005)Google Scholar
- 24.Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, Vol. I: Basic Theory. Grundlehren Math. Springer, Berlin (2006)Google Scholar