Normalized gradient-based multiobjective balanced gradient flow: Convergence analysis and acceleration

In multiobjective optimization, gradient imbalance, where objective functions have significant different gradient magnitudes, can severely slow the convergence of gradient flows. To address this issue, we propose the Multiobjective Balanced Gradient Flow (MBGF), a novel first-order dynamical system incorporating normalized gradient preprocessing techniques. Under certain assumptions, we establish the existence of the MBGF trajectory for convex multiobjective optimization problems and prove its convergence to a weakly Pareto optimal solution. By utilizing the merit functions, we further establish a sublinear convergence rate of $ O(1/t) $ for the convex case and an exponential convergence rate for the strongly convex case. For the non-convex case, we derive a convergence rate of $ O(1/\sqrt{t}) $ in terms of the minimum norm of the gradient combination. Additionally, we introduce a multiobjective accelerated scaled gradient flow and establish an improved convergence rate of $ O(1/t^2) $ for the convex case. Numerical experiments validate our theoretical results and illustrate the advantages of the proposed method.