The CP-Matrix Completion Problem

Anwa Zhou, Jinyan Fan
2014 SIAM Journal on Matrix Analysis and Applications  
A symmetric matrix C is completely positive (CP) if there exists an entrywise nonnegative matrix B such that C = BB T . The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e., a matrix having unknown entries) such that the completed matrix is completely positive. We propose a semidefinite algorithm for solving general CP-completion problems and study its properties. When all of the diagonal entries are given, the algorithm can give a
more » ... gorithm can give a certificate if a partial matrix is not CP-completable, and it almost always gives a CP-completion if it is CP-completable. When diagonal entries are partially given, similar properties hold. Computational experiments are also presented to show how CP-completion problems can be solved. Abstract A symmetric matrix A is completely positive (CP) if there exists an entrywise nonnegative matrix B such that A = B B T . We characterize the interior of the CP cone. A semidefinite algorithm is proposed for checking whether a matrix is in the interior of the CP cone, and its properties are studied. A CP-decomposition of a matrix in Dickinson's form can be obtained if it is an interior of the CP cone. Some computational experiments are also presented. Abstract A real n × n symmetric matrix P is partially positive (PP) for a given index set I ⊆ {1, . . . , n} if there exists a matrix V such that V (I, :) 0 and P = V V T . We give a characterization of PP-matrices. A semidefinite algorithm is presented for checking whether a matrix is partially positive or not. Its properties are studied. A PP-decomposition of a matrix can also be obtained if it is partially positive. Abstract. A symmetric matrix In this paper, we study the CP-matrix approximation problem: for a given symmetric matrix C, find a CP matrix X, such that X is close to C as much as possible, under some linear constraints. We formulate the problem as a linear optimization problem with the norm cone and the cone of moments, then construct a hierarchy of semidefinite relaxations for solving it. Abstract In this paper, we consider the problem of computing the distance between the linear matrix pencil and the completely positive cone. We formulate it as a linear optimization problem with the cone of moments and the second order cone. A semidefinite relaxation algorithm is presented and the convergence is studied. We also propose a new model for checking the membership in the completely positive cone. Keywords Completely positive matrices · CP projection · Linear matrix pencil · Linear optimization with moments · Semidefinite algorithm Mathematics Subject Classification Primary 15A22 · 15A23 · 44A60 · 90C22 · 90C26 · 90C59 B Jinyan Fan Abstract. A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. In this paper, we characterize the complete positive tensor as a truncated moment sequence, and transform the problem of checking whether a tensor is complete positive to checking whether its corresponding truncated moment sequence admits a representing measure, then apply Nie's method in [J. Nie, The A-truncated K-moment problem, Foundations of Computational Mathematics, 14 (2014), pp. 1243-1276] to solve it. If a tensor is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained. Abstract. This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre's hierarchy of semidefinite relaxations. We show that it has finite convergence for general tensors. Numerical experiments are presented to show the efficiency of proposed methods. a b s t r a c t Based on the ideas of LIAM and the U-support vector machine, this paper proposes a new semi-supervised proximal support vector machine, which only requires solving the inverse of an n þ 1-by-n þ 1 matrix to obtain the final classification hyperplane just as the PLIAM and is faster and more efficient than other mathematical programming-based methods. The most essential is that this method overcomes the two fundamental drawbacks of the general LIAM semi-supervised support vector classifiers: (1) they included the whole information provided by both the positively and negatively labeled instances through the unlabeled instances that are in its neighborhood in the linear constraints, which greatly added the number of constraints and made the optimization solver more complex; (2) they can only utilize the unlabeled points that are in the neighborhood of a labeled point, which may influence the accuracy of classification. The experiments on public benchmarks indicate that our semi-supervised PSVM classifier is more accurate than the original PLIAM semi-supervised classification method. Abstract In this paper, three new hybrid nonlinear conjugate gradient methods are presented, which produce sufficient descent search direction at every iteration. This property is independent of any line search or the convexity of the objective function used. Under suitable conditions, we prove that the proposed methods converge globally for general nonconvex functions. The numerical results show that all these three new hybrid methods are efficient for the given test problems. Abstract In this paper, a new nonlinear conjugate gradient method is proposed for large-scale unconstrained optimization. The sufficient descent property holds without any line searches. We use some steplength technique which ensures the Zoutendijk condition to be held, this method is proved to be globally convergent. Finally, we improve it, and do further analysis.
doi:10.1137/130919490 fatcat:tywrxsygnjf47fhqnevv5ovvie