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 the diagonal entries are given, the algorithm 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.2000 Mathematics Subject Classification. Primary 15A23, 15A48, 15A83, 90C22.
A symmetric tensor, which has a symmetric nonnegative decomposition, is called a completely positive tensor. We consider the completely positive tensor decomposition problem. A semidefinite algorithm is presented for checking whether a symmetric tensor is completely positive. If it is not completely positive, a certificate for it can be obtained; if it is completely positive, a nonnegative decomposition can be obtained.2000 Mathematics Subject Classification. Primary 15A18, 15A69, 90C22.
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.
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