We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of an entropy game, in which the state space consists of all switching sequences of a given length. We show that, by increasing this length, we arrive at a convergent approximation scheme to compute the joint spectral radius. The complexity of this method is exponential in the length of the switching sequences, but it is quite insensitive to the size of the matrices, allowing us to solve very large scale instances (several matrices in dimensions of order 1000 within a minute). An idea of this method is to replace a hierarchy of optimization problems, introduced by Ahmadi, Jungers, Parrilo and Roozbehani, by a hierarchy of nonlinear eigenproblems. To solve the latter eigenproblems, we introduce a projective version of Krasnoselskii-Mann iteration. This method is of independent interest as it applies more generally to the nonlinear eigenproblem for a monotone positively homogeneous map. Here, this method allows for scalability by avoiding the recourse to linear or semidefinite programming techniques.

中文翻译：

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非线性本征问题的收敛层次结构，用于计算非负矩阵的联合谱半径

我们证明了，一个非负矩阵的有限集合的联合谱半径可以被一个非线性算子的特征值所限制。该特征值与风险敏感的控制问题或熵博弈的遍历常数一致，在熵博弈中，状态空间由给定长度的所有切换序列组成。我们表明，通过增加该长度，我们得出了一种收敛的近似方案来计算联合光谱半径。这种方法的复杂度在切换序列的长度上成指数增长，但是它对矩阵的大小非常不敏感，从而使我们能够解决非常大规模的实例（在一分钟内有几千个维度的矩阵）。这种方法的想法是取代由Ahmadi，Jungers，Parrilo和Roozbehani引入的优化问题的层次结构，通过非线性本征问题的层次结构。为了解决后面的本征问题，我们引入了Krasnoselskii-Mann迭代的投影形式。该方法具有独立的意义，因为它更普遍地应用于单调正齐次映射的非线性本征问题。在此，该方法通过避免求助于线性或半确定编程技术来实现可伸缩性。