Gradient descent

Gradient descent is a method for finding extrema of a function by following the direction of fastest change in said function, i.e., its gradient. For energy minimization, we follow the direction corresponding to minus the gradient, that is, we update the atomic positions along the direction given by the forces. Iteratively, we propagate the atomic positions as follows: [math]\displaystyle{ \textbf{x}_{n+1} = \textbf{x}_n + \gamma_n \textbf{F}_n, \lt math\gt until we reach the criterion for convergence (set with the \lt code\gt [[e_tol]]\lt /code\gt keyword), \lt math\gt |E_{n+1} - E_n| \lt E_\text{tol} }[/math]. The [math]\displaystyle{ \gamma }[/math] parameter determines by how much we update the atomic positions along the search direction, and is allowed to change between iterations. In TurboGAP, the initial $\gamma$ is chosen by specifying the maximum allowed atomic displacement in the first optimization step, max_opt_step, followed by an automated backtracking line search.

Under construction