Inexact projected preconditioned gradient methods with variable metrics: a Lyapunov convergence theory (extended version)

Abstract

Projected gradient methods are widely used for constrained optimization. A key application is for partial differential equations (PDEs), where the objective functional represents physical energy and the linear constraints enforce conservation laws. However, computing the projections onto constraint sets generally requires solving large-scale ill-conditioned systems. A common strategy is to relax projection accuracy and apply preconditioners, which leads to inexact preconditioned projected gradient descent (IPPGD) methods studied here. Furthermore, variable preconditioners dynamically incorporating updated nonlinear information often enhance convergence rates. However, due to the complex interplay between inexactness and adaptive preconditioners, the theoretical analysis and the dynamic behavior of the IPPGD methods still remain quite open. We propose an effective strategy for constructing the inexact projection operator and develop a gradient-type flow to model the resulting IPPGD methods. Discretization of this flow not only recovers the original IPPGD method but also yields a potentially faster novel method. Furthermore, we apply Lyapunov analysis, designing a delicate Lyapunov function, to prove the exponential convergence at the continuous level and linear convergence at the discrete level under certain assumptions. Finally, we validate our approach through numerical experiments, demonstrating robust performance and computational efficiency.