In this paper, we propose a new family of continuous-time optimisation algorithms based on discontinuous second-order gradient optimisation flows, with finite-time convergence guarantees to local optima, for locally strongly convex (time-varying) cost functions. To analyse our flows, we first extend a well-know Lyapunov inequality condition for finite-time stability, to the case of (time-varying) differential inclusions. We then prove the convergence of these second-order flows in finite-time. In some particular cases, we can show that the finite-time convergence can be pre-defined by the user. We propose a robustification of the flows to bounded additive uncertainties and extend some of the results to the case of constrained optimisation. We show the performance of these flows on well-known optimisation benchmarks, namely, the Rosenbrock function, and the Rastringin function.

在本文中，我们提出了一个新的基于不连续二阶梯度优化流的连续时间优化算法系列，对于局部强凸（时变）成本函数，具有对局部最优的有限时间收敛保证。为了分析我们的流动，我们首先将有限时间稳定性的众所周知的李雅普诺夫不等式条件扩展到（时变）微分包含的情况。然后我们证明了这些二阶流在有限时间内的收敛性。在某些特定情况下，我们可以证明有限时间收敛可以由用户预先定义. 我们建议对有界附加不确定性的流动进行稳健化，并将一些结果扩展到约束优化的情况。我们在著名的优化基准（即 Rosenbrock 函数和 Rastringin 函数）上展示了这些流程的性能。