An optimally fast objective-function-free minimization algorithm using random subspaces

An algorithm for unconstrained non-convex optimization is described, which does not evaluate the objective function and in which minimization is carried out, at each iteration, within a randomly selected subspace. It is shown that this random approximation technique does not affect the method's convergence nor its evaluation complexity for the search of an $\epsilon$-approximate first-order critical point, which is $\mathcal{O}(\epsilon^{-(p+1)/p})$, where $p$ is the order of derivatives used. A variant of the algorithm using approximate Hessian matrices is also analysed and shown to require at most $\mathcal{O}(\epsilon^{-2})$ evaluations. Preliminary numerical tests show that the random-subspace technique can significantly improve performance when used with $p=2$ in the correct context, making it very competitive when compared to standard first-order algorithms.