01965nas a2200157   4500008004100000020001400041245007400055210006900129260001500198300001400213490000700227520147900234100002701713700002001740856004701760       2022                        eng d  a1573-289400aA piecewise conservative method for unconstrained convex optimization0 apiecewise conservative method for unconstrained convex optimizat  c2022/01/01  a251 - 2880 v813 aWe consider a continuous-time optimization method based on a dynamical system, where a massive particle starting at rest moves in the conservative force field generated by the objective function, without any kind of friction. We formulate a restart criterion based on the mean dissipation of the kinetic energy, and we prove a global convergence result for strongly-convex functions. Using the Symplectic Euler discretization scheme, we obtain an iterative optimization algorithm. We have considered a discrete mean dissipation restart scheme, but we have also introduced a new restart procedure based on ensuring at each iteration a decrease of the objective function greater than the one achieved by a step of the classical gradient method. For the discrete conservative algorithm, this last restart criterion is capable of guaranteeing a qualitative convergence result. We apply the same restart scheme to the Nesterov Accelerated Gradient (NAG-C), and we use this restarted NAG-C as benchmark in the numerical experiments. In the smooth convex problems considered, our method shows a faster convergence rate than the restarted NAG-C. We propose an extension of our discrete conservative algorithm to composite optimization: in the numerical tests involving non-strongly convex functions with $$\ell ^1$$-regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.1 aScagliotti, Alessandro1 aFranzone, Colli  uhttps://doi.org/10.1007/s10589-021-00332-0