We aim to factorize a completely positive matrix by using an optimization approach which consists in the minimization of a nonconvex smooth function over a convex and compact set. To solve this problem we propose a projected gradient algorithm with parameters that take into account the effects of relaxation and inertia. Both projection and gradient steps are simple in the sense that they have explicit formulas and do not require inner loops. Furthermore, no expensive procedure to find an appropriate starting point is needed. The convergence analysis shows that the whole sequence of generated iterates converges to a critical point of the objective function and it makes use of the Łojasiewicz inequality. Its rate of convergence expressed in terms of the Łojasiewicz exponent of a regularization of the objective function is also provided. Numerical experiments demonstrate the efficiency of the proposed method, in particular in comparison to other factorization algorithms, and emphasize the role of the relaxation and inertial parameters.

中文翻译：

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使用迭代投影梯度步骤对完全正矩阵进行因式分解

我们的目标是通过使用优化方法来分解完全正矩阵，该方法包括在凸集和紧集上最小化非凸平滑函数。为了解决这个问题，我们提出了一种投影梯度算法，其参数考虑了松弛和惯性的影响。投影和梯度步骤都很简单，因为它们具有明确的公式并且不需要内部循环。此外，不需要昂贵的程序来找到合适的起点。收敛分析表明生成的迭代的整个序列收敛到目标函数的一个临界点，它利用了 Łojasiewicz 不等式。还提供了以目标函数正则化的 Łojasiewicz 指数表示的收敛速度。