Finite-Time Convergence of Continuous-Time Optimization Algorithms via Differential Inclusions
This work addresses the challenge of faster convergence in optimization for researchers and practitioners, though it appears incremental as it builds on existing differential inclusion methods.
The authors tackled the problem of ensuring finite-time convergence to local minima in continuous-time optimization algorithms by proposing two discontinuous dynamical systems with provable prescribed finite-time local convergence, achieving exact settling-time bounds for strict local minima as demonstrated on the Rosenbrock function.
In this paper, we propose two discontinuous dynamical systems in continuous time with guaranteed prescribed finite-time local convergence to strict local minima of a given cost function. Our approach consists of exploiting a Lyapunov-based differential inequality for differential inclusions, which leads to finite-time stability and thus finite-time convergence with a provable bound on the settling time. In particular, for exact solutions to the aforementioned differential inequality, the settling-time bound is also exact, thus achieving prescribed finite-time convergence. We thus construct a class of discontinuous dynamical systems, of second order with respect to the cost function, that serve as continuous-time optimization algorithms with finite-time convergence and prescribed convergence time. Finally, we illustrate our results on the Rosenbrock function.