  We give quantum speedups of several general-purpose numerical optimisation
methods for minimising a function $f:\mathbb{R}^n \to \mathbb{R}$. First, we
show that many techniques for global optimisation under a Lipschitz constraint
can be accelerated near-quadratically. Second, we show that backtracking line
search, an ingredient in quasi-Newton optimisation algorithms, can be
accelerated up to quadratically. Third, we show that a component of the
Nelder-Mead algorithm can be accelerated by up to a multiplicative factor of
$O(\sqrt{n})$. Fourth, we show that a quantum gradient computation algorithm of
Gily\'en et al. can be used to approximately compute gradients in the framework
of stochastic gradient descent. In each case, our results are based on applying
existing quantum algorithms to accelerate specific components of the classical
algorithms, rather than developing new quantum techniques.
