The Iterates of the Frank–Wolfe Algorithm May Not Converge

IF 1.4 3区数学 Q2 MATHEMATICS, APPLIED

Mathematics of Operations Research Pub Date : 2023-12-07 DOI:10.1287/moor.2022.0057

Jérôme Bolte, Cyrille W. Combettes, Edouard Pauwels

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引用次数: 0

Abstract

The Frank–Wolfe algorithm is a popular method for minimizing a smooth convex function f over a compact convex set [Formula: see text]. Whereas many convergence results have been derived in terms of function values, almost nothing is known about the convergence behavior of the sequence of iterates [Formula: see text]. Under the usual assumptions, we design several counterexamples to the convergence of [Formula: see text], where f is d-time continuously differentiable, [Formula: see text], and [Formula: see text]. Our counterexamples cover the cases of open-loop, closed-loop, and line-search step-size strategies and work for any choice of the linear minimization oracle, thus demonstrating the fundamental pathologies in the convergence behavior of [Formula: see text].Funding: The authors acknowledge the support of the AI Interdisciplinary Institute ANITI funding through the French “Investments for the Future – PIA3” program under the Agence Nationale de la Recherche (ANR) agreement [Grant ANR-19-PI3A0004], the Air Force Office of Scientific Research, Air Force Material Command, U.S. Air Force [Grants FA866-22-1-7012 and ANR MaSDOL 19-CE23-0017-0], ANR Chess [Grant ANR-17-EURE-0010], ANR Regulia, and Centre Lagrange.

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弗兰克-沃尔夫算法的迭代可能不会收敛

弗兰克-沃尔夫算法是在紧凑凸集上最小化光滑凸函数 f 的常用方法[公式：见正文]。虽然许多收敛结果都是根据函数值推导出来的，但对迭代序列的收敛行为几乎一无所知[公式：见正文]。在通常的假设条件下，我们设计了几个反例来证明 f 为 d 时连续可微分的[公式：见正文]、[公式：见正文]和[公式：见正文]的收敛性。我们的反例涵盖了开环、闭环和线性搜索步长策略的情况，并且适用于任何线性最小化神谕的选择，从而证明了[公式：见正文]收敛行为的基本病理：作者感谢人工智能跨学科研究所ANITI通过法国国家研究署（ANR）协议下的 "未来投资-PIA3 "计划[赠款ANR-19-PI3A0004]、美国空军材料司令部空军科学研究办公室[赠款FA866-22-1-7012和ANR MaSDOL 19-CE23-0017-0]、ANR国际象棋[赠款ANR-17-EURE-0010]、ANR Regulia和拉格朗日中心提供的资助。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematics of Operations Research 管理科学-应用数学

CiteScore

3.40

自引率

5.90%

发文量

178

审稿时长

15.0 months

期刊介绍： Mathematics of Operations Research is an international journal of the Institute for Operations Research and the Management Sciences (INFORMS). The journal invites articles concerned with the mathematical and computational foundations in the areas of continuous, discrete, and stochastic optimization; mathematical programming; dynamic programming; stochastic processes; stochastic models; simulation methodology; control and adaptation; networks; game theory; and decision theory. Also sought are contributions to learning theory and machine learning that have special relevance to decision making, operations research, and management science. The emphasis is on originality, quality, and importance; correctness alone is not sufficient. Significant developments in operations research and management science not having substantial mathematical interest should be directed to other journals such as Management Science or Operations Research.