SIAM Journal on Optimization, Volume 31, Issue 2, Page 1079-1107, January 2021.
Structured optimization problems are ubiquitous in fields like data science and engineering. The goal in structured optimization is using a prescribed set of points, called atoms, to build up a solution that minimizes or maximizes a given function. In the present paper, we want to minimize a black-box function over the convex hull of a given set of atoms, a problem that can be used to model a number of real-world applications. We focus on problems whose solutions are sparse, i.e., solutions that can be obtained as a proper convex combination of just a few atoms in the set, and propose a suitable derivative-free inner approximation approach that nicely exploits the structure of the given problem. This enables us to properly handle the dimensionality issues usually connected with derivative-free algorithms, thus getting a method that scales well in terms of both the dimension of the problem and the number of atoms. We analyze global convergence to stationary points. Moreover, we show that, under suitable assumptions, the proposed algorithm identifies a specific subset of atoms with zero weight in the final solution after finitely many iterations. Finally, we report numerical results showing the effectiveness of the proposed method.

中文翻译：

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结构优化问题的无导数方法

SIAM优化杂志，第31卷，第2期，第1079-1107页，2021年1月。
结构优化问题在数据科学和工程学等领域无处不在。结构优化的目标是使用一组规定的点（称为原子）来构建使给定函数最小化或最大化的解决方案。在本文中，我们希望最小化给定原子集的凸包上的黑盒函数，该问题可用于对许多实际应用进行建模。我们关注解决方案稀疏的问题，即可以通过集合中几个原子的适当凸组合获得的解决方案，并提出一种合适的无导数内部近似方法，该方法很好地利用了给定问题的结构。这使我们能够正确处理通常与无导数算法有关的维数问题，从而得到一种在问题的维数和原子数方面都可以很好地扩展的方法。我们分析了全局收敛到平稳点的情况。此外，我们表明，在适当的假设下，经过有限次迭代后，所提出的算法在最终解决方案中识别出权重为零的特定原子子集。最后，我们报告了数值结果，表明了该方法的有效性。