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Structured optimization problems are ubiquitous in fields like data scienceand engineering. The goal in structured optimization is using a prescribed setof points, called atoms, to build up a solution that minimizes or maximizes agiven function. In the present paper, we want to minimize a black-box functionover the convex hull of a given set of atoms, a problem that can be used tomodel a number of real-world applications. We focus on problems whose solutionsare sparse, i.e., solutions that can be obtained as a proper convex combinationof just a few atoms in the set, and propose a suitable derivative-free innerapproximation approach that nicely exploits the structure of the given problem.This enables us to properly handle the dimensionality issues usually connectedwith derivative-free algorithms, thus getting a method that scales well interms of both the dimension of the problem and the number of atoms. We analyzeglobal convergence to stationary points. Moreover, we show that, under suitableassumptions, the proposed algorithm identifies a specific subset of atoms withzero weight in the final solution after finitely many iterations. Finally, wereport numerical results showing the effectiveness of the proposed method.

A derivative-free method for structured optimization problems