Given a symmetric nonnegative matrix A, symmetric nonnegative matrix factorization (sym-NMF) is the problem of finding a nonnegative matrix H, usually with much fewer columns than A, such that A ≈ HH T . SymNMF can be used for data analysis and in particular for various clustering tasks. In this paper, we propose simple and very efficient coordinate descent schemes to solve this problem, and that can handle large and sparse input matrices. The effectiveness of our methods is illustrated on synthetic and real-world data sets, and we show that they perform favorably compared to recent state-of-the-art methods.

This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an m-by-n nonnegative matrix X and an integer k, the PSD factorization problem consists in finding, if possible, symmetric k-by-k positive semidefinite matrices {A 1 , ..., A m } and {B 1 , ..., B n } such that Xi,j = trace(A i B j ) for i = 1, ..., m, and j = 1, ..., n. PSD factorization is NP-hard. In this work, we introduce several local optimization schemes to tackle this problem: a fast projected gradient method and two algorithms based on the coordinate descent framework. The main application of PSD factorization is the computation of semidefinite extensions, that is, the representations of polyhedrons as projections of spectrahedra, for which the matrix to be factorized is the slack matrix of the polyhedron. We compare the performance of our algorithms on this class of problems. In particular, we compute the PSD extensions of size k = 1 + log 2 (n) for the regular n-gons when n = 5, 8 and 10. We also show how to generalize our algorithms to compute the square root rank (which is the size of the factors in a PSD factorization where all factor matrices A i and B j have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality A i = B i is required for all i).

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an m-by-n nonnegative matrix X and a factorization rank r, find, if possible, an m-by-r nonnegative matrix W and an r-by-n nonnegative matrix H such that X = W H. In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive search procedure. We show that these two heuristics are able to compute exact nonnegative factorizations for several classes of nonnegative matrices (namely, linear Euclidean distance matrices, slack matrices, unique-disjointness matrices, and randomly generated matrices) and as such demonstrate their superiority over standard multi-start strategies. We also consider a hybridization between these two heuristics that allows us to combine the advantages of both methods. Finally, we discuss the use of these heuristics to gain insight on the behavior of the nonnegative rank, i.e., the minimum factorization rank such that an exact NMF exists. In particular, we disprove a conjecture on the nonnegative rank of a Kronecker product, propose a new upper bound on the extension complexity of generic n-gons and conjecture the exact value of (i) the extension complexity of regular n-gons and (ii) the nonnegative rank of a submatrix of the slack matrix of the correlation polytope.

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a b s t r a c tThis paper addresses two different problems coming from the printing industry: the label printing problem and the cover printing problem. In both cases, the problem consists in the assignment of a fixed number of labels on different templates in order to fulfill the printing requirements and to minimize the waste of excessive printed labels in the first case or to minimize the total production cost in the second case. In these problems, allocating suitable labels to each template is a strategical task to improve the efficiency of the printing process. As the considered problems are hard, we propose a new GRASP algorithm to solve these problems. The proposed algorithm is tested on some reference instances, comparing the obtained results with those found in the literature. The results prove that the GRASP algorithm is particularly well suited to these problems.