# A piecewise conservative method for unconstrained convex optimization

arXiv: https://arxiv.org/abs/2009.11233 · HTML: https://mlsanity.com/paper/2009.11233/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2020.

## Built on by (most cited first, 1 of 1)

- [Losing momentum in continuous-time stochastic optimisation](https://mlsanity.com/paper/2209.03705/) (2022)

## Abstract (arXiv)

We 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 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 ℓ1-regularization, it has better performances than the well known efficient Fast Iterative Shrinkage-Thresholding Algorithm, accelerated with an adaptive restart scheme.