Newton-like methods for solving vector optimization problems

Abstract

In the context of Euclidean spaces, we present an extension of the Newton-like method for solving vector optimization problems, with respect to the partial orders induced by a pointed, closed and convex cone with a nonempty interior. We study both exact and inexact versions of the Newton-like method. Under reasonable hypotheses, we prove stationarity of accumulation points of the sequences produced by Newton-like methods. Moreover, assuming strict cone-convexity of the objective map to the vector optimization problem, we establish convergence of the sequences to an efficient point whenever the initial point is in a compact level set.

Keywords:

• vector/multiobjective optimization
• critical point
• Newton-like method
• convergence
• scalarization

AMS Subject Classifications:

• 90C29
• 90C30

Acknowledgements

This research was partially supported by the National Natural Science Foundation of China (Grant number: 11301567) and the Fundamental Research Funds for the Central Universities (Grant number: CQDXWL-2012-010).