Simple Complexity Analysis of Simplified Direct Search

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 $\epsilon$.