  We describe a family of descent algorithms which generalizes common existing
schemes used in applications such as neural network training and more broadly
for optimization of smooth functions--potentially for global optimization, or
as a local optimization method to be deployed within global optimization
schemes like basin hopping. By introducing an auxiliary degree of freedom we
create a dynamical system with improved stability, reducing oscillatory modes
and accelerating convergence to minima. The resulting algorithms are simple to
implement and control, and convergence can be shown directly by Lyapunov's
second method.
  Although this framework, which we refer to as friction-adaptive descent
(FAD), is fairly general, we focus most of our attention here on a specific
variant: kinetic energy stabilization (which can be viewed as a
zero-temperature Nos\'e--Hoover scheme but with added dissipation in both
physical and auxiliary variables), termed KFAD (kinetic FAD). To illustrate the
flexibility of the FAD framework we consider several other methods. in certain
asymptotic limits, these methods can be viewed as introducing cubic damping in
various forms; they can be more efficient than linearly dissipated Hamiltonian
dynamics in common optimization settings.
  We present details of the numerical methods and show convergence for both the
continuous and discretized dynamics in the convex setting by constructing
Lyapunov functions. The methods are tested using a toy model (the Rosenbrock
function). We also demonstrate the methods for structural optimization for
atomic clusters in Lennard--Jones and Morse potentials. The experiments show
the relative efficiency and robustness of FAD in comparison to linearly
dissipated Hamiltonian dynamics.
