A quadratically convergent Newton method for vector optimization

Abstract

We propose a Newton method for solving smooth unconstrained vector optimization problems under partial orders induced by general closed convex pointed cones. The method extends the one proposed by Fliege, Graña Drummond and Svaiter for multicriteria, which in turn is an extension of the classical Newton method for scalar optimization. The steplength is chosen by means of an Armijo-like rule, guaranteeing an objective value decrease at each iteration. Under standard assumptions, we establish superlinear convergence to an efficient point. Additionally, as in the scalar case, assuming Lipschitz continuity of the second derivative of the objective vector-valued function, we prove q-quadratic convergence.

Keywords:

• vector optimization
• efficient points
• convexity with respect to cones
• Newton method

Acknowledgements

The authors thank the two anonymous referees for their comments, corrections and suggestions, which improved the original version of this work. The first two authors were partially supported by FAPERJ/CNPq through PRONEX-Optimization and by CNPq through Projeto Universal 473818/2007-8 and last author was partially supported by CNPq Grant 301200/93-9(RN) and by FAPERJ/CNPq through PRONEX-Optimization.