广义几何块近端线性化方法用于多块非凸非光滑优化
Generalized Geometry Block Proximal Linearized Method for Multiblock Nonconvex and Nonsmooth Optimization
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中文总结 AI 辅助
针对多块非凸非光滑优化问题，提出广义几何块近端线性化（GGBPL）方法及其惯性版本，利用任意内积和可容许度量适应问题几何结构，建立了统一收敛框架和迭代复杂度界，并在稀疏非负矩阵分解和CP分解上展现优越性能。
中文摘要 AI 辅助
本文考虑一类在许多应用中出现的多块非凸非光滑优化问题。现有方法在标准欧几里得几何中构造近端线性化算子或其变体来解决此类问题，迫使它们的块变量更新依赖于标准内积及其诱导范数。然而，这种构造未能捕捉目标问题的几何结构，导致数值效率低下。为了克服这些缺点，我们提出了一种用于更新块变量的广义几何近端线性化算子，并基于该算子发展了广义几何块近端线性化（GGBPL）方法。与现有的近端线性化算子相比，所提出的算子允许使用任意内积和一般可容许度量来构造块代理函数，从而使GGBPL方法能够使其更新适应各种问题的几何结构。我们还引入了GGBPL的惯性版本，称为惯性GGBPL（iGGBPL）方法。我们进一步在这种广义几何下建立了一个新的统一收敛框架，在该框架内我们证明了我们的方法保证目标函数值的收敛性，建立了生成序列全局收敛到临界点，并推导了我们的方法的收敛速率。我们还建立了获得ε-稳定点的$\nmathcal{O}(\varepsilon^{-2})$迭代复杂度界。我们将我们的方法应用于两个非凸非光滑问题：具有$\ell_0$约束的稀疏非负矩阵分解和具有$\ell_0$约束的稀疏非负CP分解。数值结果表明，我们提出的方法在数值性能上优于几种最先进的方法。
英文摘要
This paper considers a class of multiblock nonconvex and nonsmooth optimization problems arising in many applications. Existing methods construct proximal linearized operators or their variants within standard Euclidean geometry to solve this class of problems, forcing their block variable updates to rely on the standard inner product and its induced norm. Nevertheless, this construction fails to capture the geometric structure of the target problem, leading to low numerical efficiency. To overcome these drawbacks, we propose a generalized geometry proximal linearized operator for updating block variables, and develop the Generalized Geometry Block Proximal Linearized (GGBPL) method based on this operator. Compared with existing proximal linearized operators, the proposed operator allows the block surrogate functions to be constructed using arbitrary inner products and general admissible metrics, thereby enabling the GGBPL method to adapt its updates to the geometric structure of various problems. We also introduce the inertial version of GGBPL, named the inertial GGBPL (iGGBPL) method. We further establish a new unified convergence framework under this generalized geometry, within which we prove that our methods guarantee convergence of the objective function values, establish global convergence of the generated sequence to a critical point, and derive the convergence rate of our methods. We also establish an $\mathcal{O}(\varepsilon^{-2})$ iteration complexity bound for obtaining an $\varepsilon$-stationary point. We apply our methods to two nonconvex and nonsmooth problems: sparse nonnegative matrix factorization with $\ell_0$-constraints and sparse nonnegative CP decomposition with $\ell_0$-constraints. Numerical results demonstrate the superior numerical performance of our proposed methods over several state-of-the-art methods.
