I have $8$ sets $E_1, \ldots, E_8$, each one of size $256$,  whose elements are $4\times 4$ matrices of (nonnegative) integers. I'd like to design an algorithm to find:$$\underset{M_1 \in E_1, \ldots, M_8 \in E_8}{\max} ||M_1 \times \ldots \times M_8||_1$$where:$$||M||_1=\sum_{i=1}^4\sum_{j=1}^4 M_{i,j}$$A reasonable time complexity might be approximately $2^{32}$ (time complexity of the naive algorithm is $2^{79}$).My first idea would be to look for interpretations in various mathematic fields:geometry: matrix product in an euclidian space.probability: it reminds me of Markov chains (but the $M_i$ are not normalized).Is there any known work about this?
Maximize entrywise 1-norm of matrix product
algorithms;markov chains