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An algorithm for unconstrained non-convex optimization is described, whichdoes not evaluate the objective function and in which minimization is carriedout, at each iteration, within a randomly selected subspace. It is shown thatthis random approximation technique does not affect the method's convergencenor its evaluation complexity for the search of an $\epsilon$-approximatefirst-order critical point, which is $\mathcal{O}(\epsilon^{-(p+1)/p})$, where$p$ is the order of derivatives used. A variant of the algorithm usingapproximate Hessian matrices is also analyzed and shown to require at most$\mathcal{O}(\epsilon^{-2})$ evaluations. Preliminary numerical tests show thatthe random-subspace technique can significantly improve performance on someproblems, albeit, unsurprisingly, not for all.

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