Robustness in Deterministic Vector Optimization
In this paper, robust efficient solutions of a vector optimization problem, whose image space is ordered by an arbitrary ordering cone, are defined. This is done from different points of view, including set based and norm based. The relationships between these solution concepts are established. Furthermore, it is shown that, for a general vector optimization problem, each norm-based robust efficient solution is a strictly efficient solution; each isolated efficient solution is a norm-based robust efficient solution; and, under appropriate assumptions, each norm-based robust efficient solution is a Henig properly efficient solution. Various necessary and sufficient conditions for characterizing norm-based robust solutions of a general vector optimization problem, in terms of the tangent and normal cones and the nonascent directions, are presented. An optimization problem for calculating a robustness radius is provided, and then, the largest robustness radius is determined. Moreover, some alterations of objective functions preserving weak/strict/Henig proper/robust efficiency are studied.
KeywordsVector optimization Nonsmooth optimization Robust solutions Robustness radius
Mathematics Subject Classification90C29 90C31 49J52
The authors would like to express their gratitude to the editor in chief of JOTA, handling editor, and two anonymous referees for their helpful comments on the earlier versions of the paper. The research of the second author was in part supported by a grant from the Iran National Science Foundation (INSF) (No. 95849588).
- 8.Bokrantz, R., Fredriksson, A.: On solutions to robust multiobjective optimization problems that are optimal under convex scalarization. arXiv:1308.4616v2 (2013)
- 24.Kuhn, H., Tucker, A.: Proceedings of the second Berkeley symposium on mathematical statistics and probability. In: Neyman, J. (ed.) Nonlinear Programming, pp. 481–492. University of California Press, Berkeley (1951)Google Scholar
- 26.Markowitz, H.: Portfolio selection. J. Finance 7(1), 77–91 (1952)Google Scholar
- 27.Markowitz, H.M.: Portfolio Selection-efficient Diversification Of Investments. Wiley, New York (1959)Google Scholar
- 31.Franklin, J.N.: Methods of Mathematical Economics: Linear and Nonlinear Programming. Cambridge University Press, Cambridge (2003)Google Scholar
- 35.Rockafellar, R.T.: Convex Analysis, 2nd edn. Princeton University, Princeton (1972)Google Scholar