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

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.