Analysis of the Frank-Wolfe method for convex composite optimization involving a logarithmically-homogeneous barrier

• 成果类型: Article
• 署名作者: Zhao, Renbo; Freund, Robert M.
• 署名单位: Massachusetts Institute of Technology (MIT); Massachusetts Institute of Technology (MIT)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01820-9
• 发表日期: 2023
• 页码: 123-163
• 关键词: conditional gradient algorithms; 1st-order methods; CONVERGENCE; inverse; MODEL; rates

摘要

We present and analyze a new generalized Frank-Wolfe method for the composite optimization problem (P) : min(x is an element of)(Rn) f (Ax) + h (x), where f is a theta-logarithmically-homogeneous self-concordant barrier, A is a linear operator and the function h has a bounded domain but is possibly non-smooth. We show that our generalized Frank-Wolfe method requires O((delta(0) + theta + R-h) ln(delta(0)) + (theta + R-h)(2)/epsilon) iterations to produce an epsilon-approximate solution, where delta(0) denotes the initial optimality gap and R-h is the variation of h on its domain. This result establishes certain intrinsic connections between theta-logarithmically homogeneous barriers and the Frank-Wolfe method. When specialized to the D-optimal design problem, we essentially recover the complexity obtained by Khachiyan (Math Oper Res 21 (2): 307-320, 1996) using the Frank-Wolfe method with exact line-search. We also study the (Fenchel) dual problem of (P), and we show that our new method is equivalent to an adaptive-step-size mirror descent method applied to the dual problem. This enables us to provide iteration complexity bounds for the mirror descent method despite the fact that the dual objective function is non-Lipschitz and has unbounded domain. In addition, we present computational experiments that point to the potential usefulness of our generalized Frank-Wolfe method on Poisson image de-blurring problems with TV regularization, and on simulated PET problem instances.