Extended Newton Methods for Multiobjective Optimization: Majorizing Function Technique and Convergence Analysis

We consider the extended Newton method for approaching a Pareto optimum of a multiobjective optimization problem, establish quadratic convergence criteria, and estimate a radius of convergence ball under the assumption that the Hessians of objective functions satisfy an L-average Lipschitz condition. These convergence theorems significantly improve the corresponding ones in [J. Fliege, L. M. G. Drummond, and B. F. Svaiter, SIAM J. Optim., 20 (2009), pp. 602--626]. As applications of the obtained results, convergence theorems under the classical Lipschitz condition or the \gamma -condition are presented for multiobjective optimization, and the global quadratic convergence
results of the extended Newton method with Armijo/Goldstein/Wolfe line-search schemes are also provided.