On the convergence of an improved and adaptive kinetic simulated annealing

Abstract

Inspired by the work of [Fang et al.. An improved annealing method and its large-time behaviour. Stochastic Process. Appl. (1997), Volume 71 Issue 1 Page 55-74.], who propose an improved simulated annealing algorithm based on a variant of overdamped Langevin diffusion with state-dependent diffusion coefficient, we cast this idea in the kinetic setting and develop an improved kinetic simulated annealing (IKSA) method for minimizing a target function $U$. To analyze its convergence, we utilize the framework recently introduced by [Monmarch\'{e}. Hypocoercivity in metastable settings and kinetic simulated annealing. Probab. Theory Related Fields (2018), Volume 172 Page 1215-1248.] for the case of kinetic simulated annealing (KSA). The core idea of IKSA rests on introducing a parameter $c > \inf U$, which de facto modifies the optimization landscape and clips the critical height in IKSA at a maximum of $c - \inf U$. Consequently IKSA enjoys improved convergence with faster logarithmic cooling than KSA. To tune the parameter $c$, we propose an adaptive method that we call IAKSA which utilizes the running minimum generated by the algorithm on the fly, thus avoiding the need to manually adjust $c$ for better performance. We present positive numerical results on some standard global optimization benchmark functions that verify the improved convergence of IAKSA over other Langevin-based annealing methods.