TRFD: A derivative-free trust-region method based on finite differences for composite nonsmooth optimization

In this work we present TRFD, a derivative-free trust-region method based on finite differences for minimizing composite functions of the form $f(x)=h(F(x))$, where $F$ is a black-box function assumed to have a Lipschitz continuous Jacobian, and $h$ is a known convex Lipschitz function, possibly nonsmooth. The method approximates the Jacobian of $F$ via forward finite differences. We establish an upper bound for the number of evaluations of $F$ that TRFD requires to find an $\epsilon$-approximate stationary point. For L1 and Minimax problems, we show that our complexity bound reduces to $\mathcal{O}(n\epsilon^{-2})$ for specific instances of TRFD, where $n$ is the number of variables of the problem. Assuming that $h$ is monotone and that the components of $F$ are convex, we also establish a worst-case complexity bound, which reduces to $\mathcal{O}(n\epsilon^{-1})$ for Minimax problems. Numerical results are provided to illustrate the relative efficiency of TRFD in comparison with existing derivative-free solvers for composite nonsmooth optimization.