  We provide a simple and generic adaptive restart scheme for convex
optimization that is able to achieve worst-case bounds matching (up to constant
multiplicative factors) optimal restart schemes that require knowledge of
problem specific constants. The scheme triggers restarts whenever there is
sufficient reduction of a distance-based potential function. This potential
function is always computable.
  We apply the scheme to obtain the first adaptive restart algorithm for
saddle-point algorithms including primal-dual hybrid gradient (PDHG) and
extragradient. The method improves the worst-case bounds of PDHG on bilinear
games, and numerical experiments on quadratic assignment problems and matrix
games demonstrate dramatic improvements for obtaining high-accuracy solutions.
Additionally, for accelerated gradient descent (AGD), this scheme obtains a
worst-case bound within 60% of the bound achieved by the (unknown) optimal
restart period when high accuracy is desired. In practice, the scheme is
competitive with the heuristic of O'Donoghue and Candes (2015).

Optimization and Control