Complexity bound of trust-region methods for convex smooth unconstrained multiobjective optimization

In this paper, we analyze the worst-case complexity of trust-region methods for solving unconstrained smooth multiobjective optimization problems. We particularly focus on the method proposed by Villacorta et al. [J Optim Theory Appl 160:865–889, 2014]. When the component functions are convex (respectively strongly convex), we will derive a complexity bound of O(ϵ-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(\epsilon ^{-1})$$\end{document} (respectively O(logϵ-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}}(\log \epsilon ^{-1})$$\end{document}) for driving some criticality measure below some given positive ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon$$\end{document}. The derived complexity bounds recover those of classical trust-region methods for solving (strongly) convex smooth unconstrained single-objective problems.