A Finite-Time Convergent Primal-Dual Gradient Dynamics Based on the Multivariable Super-Twisting Algorithm

We propose a novel primal-dual gradient dynamics (PDGD) algorithm to dynamically solve an optimization problem with linear equality constraints in finite time. To ensure finite-time convergence, we endow the PDGD with suitable homogeneity properties. More precisely, departing from the standard PDGD and based on the associated Lagrangian of the optimization problem, the algorithm is derived by suitably combining a change of coordinates of the standard PDGD with the multivariable super-twisting algorithm. In our new coordinates, the proposed PDGD's global convergence to the optimal solution of the optimization problem is then proven via a smooth, strong Lyapunov function. Additionally, we provide a numerical example to compare the performance of our algorithm with existing approaches from the literature.