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Abstract. The polyhedral homotopy method, which has been known as a powerful numerical method for computing all isolated zeros of a polynomial system, requires all mixed cells of the support of the system to construct a family of homotopy functions. Finding the mixed cells is formulated in terms of a linear inequality system with an additional combinatorial condition. It is essential in computational efficiency how we construct an enumeration tree among a family of linear inequalities induced from it such that every mixed cell corresponds to a unique feasible leaf node. This paper proposes a dynamic construction of an enumeration tree, which branches each parent node into its child nodes so that the number of feasible child nodes is expected to be small; hence we can prune a lot of subtrees which do not contain any mixed cell. Numerical results exhibit that our dynamic construction of an enumeration tree works very efficiently for large scale polynomial systems; for example, it generated all mixed cells of the cyclic-15 problem for the first time in less than 16 hours.

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The successive projection algorithm (SPA) has been known to work well for separable nonnegative matrix factorization (NMF) problems arising in applications, such as topic extraction from documents and endmember detection in hyperspectral images. One of the reasons is in that the algorithm is robust to noise. Gillis and Vavasis showed in [SIAM J. Optim., 25(1), pp. 677-698, 2015] that a preconditioner can further enhance its noise robustness. The proof rested on the condition that the dimension d and factorization rank r in the separable NMF problem coincide with each other. However, it may be unrealistic to expect that the condition holds in separable NMF problems appearing in actual applications; in such problems, d is usually greater than r. This paper shows, without the condition d = r, that the preconditioned SPA is robust to noise.

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