A Finite-Difference Trust-Region Method for Convexly Constrained Smooth Optimization

We propose a derivative-free trust-region method based on finite-difference gradient approximations for smooth optimization problems with convex constraints. The proposed method does not require computing an approximate stationarity measure. For nonconvex problems, we establish a worst-case complexity bound of $\mathcal{O}\text{\hspace{0.17em}⁣}\left(n{\left(\frac{L}{\sigma }\epsilon \right)}^{-2}\right)$ function evaluations for the method to reach an $\left(\frac{L}{\sigma }\epsilon \right)$-approximate stationary point, where $n$ is the number of variables, $L$ is the Lipschitz constant of the gradient, and $\sigma$ is a user-defined estimate of $L$. If the objective function is convex, the complexity to reduce the functional residual below $(L\mathrm{/}\sigma )\epsilon$ is shown to be of $\mathcal{O}\text{\hspace{0.17em}⁣}\left(n{\left(\frac{L}{\sigma }\epsilon \right)}^{-1}\right)$ function evaluations, while for Polyak-Lojasiewicz functions on unconstrained domains, the bound further improves to $\mathcal{O}(n\log \left({\left(\frac{L}{\sigma }\epsilon \right)}^{-1}\right))$. Numerical experiments on benchmark problems and a model-fitting application demonstrate the method’s efficiency relative to state-of-the-art derivative-free solvers for both unconstrained and bound-constrained problems.