Optimal schedules for annealing algorithms

Annealing algorithms such as simulated annealing and population annealing are widely used both for sampling the Gibbs distribution and solving optimization problems (i.e., finding ground states). For both statistical mechanics and optimization, additional parameters beyond temperature are often needed such as chemical potentials, external fields, or Lagrange multipliers enforcing constraints. In this paper we derive a formalism for optimal annealing schedules in multidimensional parameter spaces using methods from nonequilibrium statistical mechanics. The results are closely related to work on optimal control of thermodynamic systems [Sivak and Crooks, Phys. Rev. Lett. 108, 190602 (2012)0031-900710.1103/PhysRevLett.108.190602]. Within the formalism, we compare the efficiency of population annealing and multiple weighted runs of simulated annealing ("annealed importance sampling") and discuss the effects of nonergodicity on both algorithms. Theoretical results are supported by numerical simulations of spin glasses.