Projection onto hyperbolicity cones and beyond: a dual Frank-Wolfe approach

• 成果类型: Article; Early Access
• 署名作者: Nagano, Takayuki; Lourenco, Bruno F.; Takeda, Akiko
• 署名单位: University of Tokyo; Research Organization of Information & Systems (ROIS); Institute of Statistical Mathematics (ISM) - Japan; RIKEN
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-026-02371-z
• 发表日期: 2026-06-19
• 关键词: Polynomials; algorithms; INEQUALITY; operator

摘要

We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we propose a Frank-Wolfe method to handle this task and, more generally, strongly convex optimization over closed convex cones. One of our innovations is that the Frank-Wolfe method is actually applied to the dual problem and, by doing so, subproblems can be solved in closed-form using minimum eigenvalue functions and conjugate vectors. To test the validity of our proposed approach, we present numerical experiments where we check the performance of alternative approaches including interior point methods and an earlier accelerated gradient method proposed by Renegar. We also show numerical examples where the hyperbolic polynomial has millions of monomials. Finally, we also discuss the problem of projecting onto p-cones which, although not hyperbolicity cones in general, are still amenable to our techniques.