A Derivative-Free CoMirror Algorithm

We consider $\min\{f(x):g(x) \le 0, ~x\in X\},$ where $X$ is a compact convex subset of $\RR^m$, and $f$ and $g$ are continuous convex functions defined on an open neighbourhood of $X$. We work in the setting of derivative-free optimization, assuming that $f$ and $g$ are available through a black-box that provides only function values for a lower-$\mathcal{C}^2$ representation of the functions. We present a derivative-free optimization variant of the $\eps$-comirror algorithm \cite{BBTGBT2010}. Algorithmic convergence hinges on the ability to accurately approximate subgradients of lower-$\mathcal{C}^2$ functions, which we prove is possible through linear interpolation. We provide convergence analysis that quantifies the difference between the function values of the iterates and the optimal function value. We find that the DFO algorithm we develop has the same convergence result as the original gradient-based algorithm. We present some numerical testing that demonstrate the practical feasibility of the algorithm, and conclude with some directions for further research.