A perturbed preconditioned gradient descent method for the unconstrained minimization of composite objectives

2025/12/22 by Park, Jea-Hyun, Salgado, Abner J., Wise, Steven M.
#FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2512.19532

Abstract

We introduce a perturbed preconditioned gradient descent (PPGD) method for the unconstrained minimization of a strongly convex objective G with a locally Lipschitz continuous gradient. We assume that G(v)=E(v)+F(v) and that the gradient of F is only known approximately. Our analysis is conducted in infinite dimensions with a preconditioner built into the framework. We prove a linear rate of convergence, up to an error term dependent on the gradient approximation. We apply the PPGD to the stationary Cahn-Hilliard equations with variable mobility under periodic boundary conditions. Numerical experiments are presented to validate the theoretical convergence rates and explore how the mobility affects the computation.

Citations

• Preconditioned accelerated gradient descent methods for locally\n Lipschitz smooth objectives with applications to the solution of nonlinear\n PDEs
• First-order methods of smooth convex optimization with inexact oracle
• Convergence Rates of Inexact Proximal-Gradient Methods for Convex Optimization
• Smooth Optimization with Approximate Gradient
• An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
• Splitting Algorithms for the Sum of Two Nonlinear Operators

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