Math. Program., Ser. A https://doi.org/10.1007/s10107-018-1285-1 FULL LENGTH PAPER Quadratic optimization with orthogonality constraint: explicit Łojasiewicz exponent and linear convergence of retraction-based line-search and stochastic variance-reduced gradient methods 1 1,2 Huikang Liu · Anthony Man-Cho So · Weijie Wu Received: 17 November 2017 / Accepted: 30 April 2018 © Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018 Abstract The problem of optimizing a quadratic form over an orthogonality constraint (QP-OC for short) is one of the most fundamental matrix optimization problems and arises in many applications. In this paper, we characterize the growth behavior of the objective function around the critical points of the QP-OC problem and demonstrate how such characterization can be used to obtain strong convergence rate results for iterative methods that exploit the manifold structure of the orthogonality constraint (i.e., the Stiefel manifold) to ﬁnd a critical point of the problem. Speciﬁcally, our primary contribution is to show that the Łojasiewicz exponent at any critical point of the QP-OC problem is 1/2. Such a result is signiﬁcant, as it expands the currently very limited repertoire of optimization problems for which the Łojasiewicz exponent is explicitly known. Moreover, it allows us to show, in
Mathematical Programming – Springer Journals
Published: Jun 1, 2018
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