Multiobjective Balanced Gradient Flow: A Dynamical Perspective on a Class of Optimization Algorithms

This paper proposes a novel dynamical system called the Multiobjective Balanced Gradient Flow (MBGF), offering a dynamical perspective for normalized gradient methods in a class of multi-objective optimization problems. Under certain assumptions, we prove the existence of solutions for MBGF trajectories and establish their convergence to weak Pareto points in the case of convex objective functions. For both convex and non-convex scenarios, we provide convergence rates of $O(1\mathrm{/}t)$ and $O(1\mathrm{/}\sqrt{t})$, respectively.