  We consider the problem of unconstrained minimization of a smooth function in
the derivative-free setting using. In particular, we propose and study a
simplified variant of the direct search method (of direction type), which we
call simplified direct search (SDS). Unlike standard direct search methods,
which depend on a large number of parameters that need to be tuned, SDS depends
on a single scalar parameter only.
  Despite relevant research activity in direct search methods spanning several
decades, complexity guarantees---bounds on the number of function evaluations
needed to find an approximate solution---were not established until very
recently. In this paper we give a surprisingly brief and unified analysis of
SDS for nonconvex, convex and strongly convex functions. We match the existing
complexity results for direct search in their dependence on the problem
dimension ($n$) and error tolerance ($\epsilon$), but the overall bounds are
simpler, easier to interpret, and have better dependence on other problem
parameters. In particular, we show that for the set of directions formed by the
standard coordinate vectors and their negatives, the number of function
evaluations needed to find an $\epsilon$-solution is $O(n^2 /\epsilon)$ (resp.
$O(n^2 \log(1/\epsilon))$) for the problem of minimizing a convex (resp.
strongly convex) smooth function. In the nonconvex smooth case, the bound is
$O(n^2/\epsilon^2)$, with the goal being the reduction of the norm of the
gradient below $\epsilon$.
