A Simple Method for Convex Optimization in the Oracle Model

Daniel Dadush, Christopher Hojny, Sophie Huiberts, Stefan Weltge

Abstract: We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function f over a convex set K given by a separation oracle. Our method utilizes the Frank–Wolfe algorithm over the cone of valid inequalities of K and subgradients of f. Under the assumption that f is L-Lipschitz and that K contains a ball of radius r and is contained inside the origin centered ball of radius R, using \(O(\frac{(RL)^2}{\varepsilon ^2} \cdot \frac{R^2}{r^2})\) iterations and calls to the oracle, our main method outputs a point \(x \in K\) satisfying \(f(x) \le \varepsilon + \min _{z \in K} f(z)\).