Convergence and complexity guarantees for a wide class of descent algorithms in nonconvex multi-objective optimization

Abstract

We address conditions for global convergence and worst-case complexity bounds of descent algorithms in nonconvex multi-objective optimization. Specifically, we define the concept of steepest-descent-related directions. We consider iterative algorithms taking steps along such directions, selecting the stepsize according to a standard Armijo-type rule. We prove that methods fitting this framework automatically enjoy global convergence properties. Moreover, we show that a slightly stricter property, satisfied by most known algorithms, guarantees the same complexity bound of $O\left({ϵ}^{-2}\right)$ as the steepest descent method.