BDF schemes for accelerated gradient flows in projection-free approximation of nonconvex constrained variational minimization

TL;DR

This paper introduces new BDF-based algorithms that combine accelerated gradient flows with projection-free methods to efficiently solve non-convex constrained variational problems, providing universal constraint violation bounds and improved stability.

Contribution

The paper develops a general framework for analyzing constraint violations in projection-free methods and introduces a new family of BDF-k accelerated gradient algorithms with energy stability.

Findings

Universal constraint violation bounds depending on iterate regularity.

High-order constraint violation estimates for BDF-k methods.

Numerical results show superior efficiency and accuracy over existing methods.

Abstract

We propose novel algorithms combining accelerated gradient flows with linearized projection-free treatments of non-convex constraints and BDF pseudo-temporal discretization for quadratic energy minimization. A general framework is developed to analyze constraint violations in such projection-free techniques for quadratic constraints. This analysis proves to be universal to all projection-free iterative methods, and constraint error bounds depend solely on iterate regularity. For BDF-k(k=1,2,3,4), we derive both unconditional and conditional high-order constraint violation estimates for accelerated gradient flows using our framework. We further discover a new family of BDF-k accelerated gradient methods achieving modified energy stability for arbitrary positive integer k. Numerical experiments validate our theoretical results and demonstrate superior efficiency and accuracy compared to existing gradient flow approaches.