We propose an adaptive time step with energy for a large class of preconditioned gradient descent methods, mainly applied to constrained optimization problems. Our strategy relies on representing the usual descent direction by the product of an energy variable and a transformed gradient, with a preconditioning matrix, for example, to reflect the natural gradient induced by the underlying metric in parameter space or to endow a projection operator when linear equality constraints are present. We present theoretical results on both unconditional stability and convergence rates for three respective classes of objective functions. In addition, our numerical results shed light on the excellent performance of the proposed method on several benchmark optimization problems.

翻译：针对一类广泛使用的预条件梯度下降法，我们提出了一种结合能量的自适应时间步长策略，主要应用于约束优化问题。该方法将通常的下降方向表示为能量变量与变换梯度的乘积，并通过预条件矩阵（例如，反映参数空间中由底层度量诱导的自然梯度，或在存在线性等式约束时赋予投影算子）来实现。针对三类目标函数，我们分别给出了无条件和收敛率的理论分析。此外，数值结果表明，所提方法在多个基准优化问题上具有优异的性能。