We present a novel optimization algorithm, element-wise relaxed scalar auxiliary variable (E-RSAV), that satisfies an unconditional energy dissipation law and exhibits improved alignment between the modified and the original energy. Our algorithm features rigorous proofs of linear convergence in the convex setting. Furthermore, we present a simple accelerated algorithm that improves the linear convergence rate to super-linear in the univariate case. We also propose an adaptive version of E-RSAV with Steffensen step size. We validate the robustness and fast convergence of our algorithm through ample numerical experiments.

翻译：我们提出了一种新颖的优化算法——逐元素松弛标量辅助变量法（E-RSAV），该算法满足无条件能量耗散定律，并展现出修正能量与原始能量之间更好的对齐性。该算法在凸情形下具有严格的线性收敛性证明。此外，我们提出了一种简单的加速算法，可将单变量情形下的线性收敛速率提升至超线性。我们还提出了一种采用Steffensen步长的自适应E-RSAV版本。通过丰富的数值实验验证了该算法的鲁棒性与快速收敛性。