Radial duality part I: foundations

• 成果类型: Article
• 署名作者: Grimmer, Benjamin
• 署名单位: Johns Hopkins University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-02006-7
• 发表日期: 2024
• 页码: 33-68
• 关键词: gauge

摘要

Renegar (SIAM J Optim 26(4):2649-2676, https://doi.org/10.1137/15M1027371 2016) introduced a novel approach to transforming generic conic optimization problems into unconstrained, uniformly Lipschitz continuous minimization. We introduce radial transformations generalizing these ideas, equipped with an entirely new motivation and development that avoids any reliance on convex cones or functions. Of practical importance, this facilitates the development of new families of projection-free first-order methods applicable even in the presence of nonconvex objectives and constraint sets. Our generalized construction of this radial transformation uncovers that it is dual (i.e., self-inverse) for a wide range of functions including all concave objectives. This gives a new duality relating optimization problems to their radially dual problem. For a broad class of functions, we characterize continuity, differentiability, and convexity under the radial transformation as well as develop a calculus for it. This radial duality provides a foundation for designing projection-free radial optimization algorithms, which is carried out in the second part of this work.