A simple method for convex optimization in the oracle model

• 成果类型: Article
• 署名作者: Dadush, Daniel; Hojny, Christopher; Huiberts, Sophie; Weltge, Stefan
• 署名单位: Eindhoven University of Technology; Columbia University; Technical University of Munich
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-02005-8
• 发表日期: 2024
• 页码: 283-304
• 关键词: approximation algorithms; fractional packing; perceptron; FLOW

摘要

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 L)2 ) R, using O((RL)(2)/(e)2 . R-2/ r(2)) iterations and calls to the oracle, our main method outputs a point x ? K satisfying f (x) = e +min(z?K) f (z). Our algorithm is easy to implement, and we believe it can serve as a useful alternative to existing cutting plane methods. As evidence towards this, we show that it compares favorably in terms of iteration counts to the standard LP based cutting plane method and the analytic center cutting plane method, on a testbed of combinatorial, semidefinite and machine learning instances.