We consider an important problem in scientific discovery, namely identifying sparse governing equations for nonlinear dynamical systems. This involves solving sparse ridge regression problems to provable optimality in order to determine which terms drive the underlying dynamics. We propose a fast algorithm, OKRidge, for sparse ridge regression, using a novel lower bound calculation involving, first, a saddle point formulation, and from there, either solving (i) a linear system or (ii) using an ADMM-based approach, where the proximal operators can be efficiently evaluated by solving another linear system and an isotonic regression problem. We also propose a method to warm-start our solver, which leverages a beam search. Experimentally, our methods attain provable optimality with run times that are orders of magnitude faster than those of the existing MIP formulations solved by the commercial solver Gurobi.

翻译：我们研究了科学发现中的一个重要问题，即识别非线性动力系统的稀疏控制方程。这需要通过求解稀疏岭回归问题以达到可证明的最优性，从而确定驱动底层动力学的关键项。我们提出了一种快速算法OKRidge用于稀疏岭回归，该算法采用了一种新颖的下界计算方法：首先构建鞍点公式，随后通过求解（i）线性系统或（ii）基于ADMM的方法进行求解——其中近端算子可通过求解另一个线性系统和一个保序回归问题高效评估。我们还提出了一种利用束搜索进行热启动求解器的方法。实验结果表明，我们的方法在实现可证明最优性的同时，其运行速度比现有由商业求解器Gurobi求解的MIP公式快数个数量级。