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\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . In this paper, we study active set identification results for the away-step Frank--Wolfe algorithm in different settings. We first prove a local identification property that we apply, in combination with a convergence hypothesis, to get an active set identification result. We then prove, for nonconvex objectives, a novel O(1/ \surd k) convergence rate result and active set identification for different step sizes (under suitable assumptions on the set of stationary points). By exploiting those results, we also give explicit active set complexity bounds for both strongly convex and nonconvex objectives. While we initially consider the probability simplex as feasible set, in an appendix we show how to adapt some of our results to generic polytopes.\bfK \bfe \bfy \bfw \bfo \bfr \bfd \bfs . surface identification, manifold identification, active set complexity \bfA \bfM \bfS \bfs \bfu \bfb \bfj \bfe \bfc \bft \bfc \bfl \bfa \bfs \bfs \bfi fi\bfc \bfa \bft \bfi \bfo \bfn \bfs . 65K05, 90C06, 90C30 \bfD \bfO \bfI .

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Invented some 65 years ago in a seminal paper by Marguerite Straus-Frank and Philip Wolfe, the Frank–Wolfe method recently enjoys a remarkable revival, fuelled by the need of fast and reliable first-order optimization methods in Data Science and other relevant application areas. This review tries to explain the success of this approach by illustrating versatility and applicability in a wide range of contexts, combined with an account on recent progress in variants, improving on both the speed and efficiency of this surprisingly simple principle of first-order optimization.

Cluster detection plays a fundamental role in the analysis of data. In this paper, we focus on the use of sdefective clique models for network-based cluster detection and propose a nonlinear optimization approach that efficiently handles those models in practice. In particular, we introduce an equivalent continuous formulation for the problem under analysis, and we analyze some tailored variants of the Frank-Wolfe algorithm that enable us to quickly find maximal s-defective cliques. The good practical behavior of those algorithmic tools, which is closely connected to their support identification properties, makes them very appealing in practical applications. The reported numerical results clearly show the effectiveness of the proposed approach.