  We consider unconstrained randomized optimization of smooth convex objective
functions in the gradient-free setting. We analyze Random Pursuit (RP)
algorithms with fixed (F-RP) and variable metric (V-RP). The algorithms only
use zeroth-order information about the objective function and compute an
approximate solution by repeated optimization over randomly chosen
one-dimensional subspaces. The distribution of search directions is dictated by
the chosen metric.
  Variable Metric RP uses novel variants of a randomized zeroth-order Hessian
approximation scheme recently introduced by Leventhal and Lewis (D. Leventhal
and A. S. Lewis., Optimization 60(3), 329--245, 2011). We here present (i) a
refined analysis of the expected single step progress of RP algorithms and
their global convergence on (strictly) convex functions and (ii) novel
convergence bounds for V-RP on strongly convex functions. We also quantify how
well the employed metric needs to match the local geometry of the function in
order for the RP algorithms to converge with the best possible rate.
  Our theoretical results are accompanied by numerical experiments, comparing
V-RP with the derivative-free schemes CMA-ES, Implicit Filtering, Nelder-Mead,
NEWUOA, Pattern-Search and Nesterov's gradient-free algorithms.

Optimization and Control