Simplex Frank-Wolfe: Linear Convergence and Its Numerical Efficiency for Convex Optimization over Polytopes

Abstract: We investigate variants of the Frank-Wolfe (FW) algorithm for smoothing and strongly convex optimization over polyhedral sets, with the goal of designing algorithms that achieve linear convergence while minimizing per-iteration complexity as much as possible. Starting from the simple yet fundamental unit simplex, and based on geometrically intuitive motivations, we introduce a novel oracle called Simplex Linear Minimization Oracle (SLMO), which can be implemented with the same complexity as the standard FW oracle. We then present two FW variants based on SLMO: Simplex Frank-Wolfe and the refined Simplex Frank-Wolfe (rSFW). Both variants achieve a linear convergence rate for all three common step-size rules. Finally, we generalize the entire framework from the unit simplex to arbitrary polytopes. Furthermore, the refinement step in rSFW can accommodate any existing FW strategies such as the well-known away-step and pairwise-step, leading to outstanding numerical performance. We emphasize that the oracle used in our rSFW method requires only one more vector addition compared to the standard LMO, resulting in the lowest per-iteration computational overhead among all known Frank-Wolfe variants with linear convergence.