202 research outputs found
Positive Definite Tensors to Nonlinear Complementarity Problems
The main purpose of this note is to investigate some kinds of nonlinear
complementarity problems (NCP). For the structured tensors, such as, symmetric
positive definite tensors and copositive tensors, we derive the existence
theorems on a solution of these kinds of nonlinear complementarity problems. We
prove that a unique solution of the NCP exists under the condition of
diagonalizable tensors.Comment: 11 page
Reduced Complexity Filtering with Stochastic Dominance Bounds: A Convex Optimization Approach
This paper uses stochastic dominance principles to construct upper and lower
sample path bounds for Hidden Markov Model (HMM) filters. Given a HMM, by using
convex optimization methods for nuclear norm minimization with copositive
constraints, we construct low rank stochastic marices so that the optimal
filters using these matrices provably lower and upper bound (with respect to a
partially ordered set) the true filtered distribution at each time instant.
Since these matrices are low rank (say R), the computational cost of evaluating
the filtering bounds is O(XR) instead of O(X2). A Monte-Carlo importance
sampling filter is presented that exploits these upper and lower bounds to
estimate the optimal posterior. Finally, using the Dobrushin coefficient,
explicit bounds are given on the variational norm between the true posterior
and the upper and lower bounds
The necessary and sufficient conditions of copositive tensors
In this paper, it is proved that (strict) copositivity of a symmetric tensor
is equivalent to the fact that every principal sub-tensor of
has no a (non-positive) negative -eigenvalue. The
necessary and sufficient conditions are also given in terms of the
-eigenvalue of the principal sub-tensor of the given tensor. This
presents a method of testing (strict) copositivity of a symmetric tensor by
means of the lower dimensional tensors. Also the equivalent definition of
strictly copositive tensors is given on entire space .Comment: 13 pages. arXiv admin note: text overlap with arXiv:1302.608
New approximations for the cone of copositive matrices and its dual
We provide convergent hierarchies for the cone C of copositive matrices and
its dual, the cone of completely positive matrices. In both cases the
corresponding hierarchy consists of nested spectrahedra and provide outer
(resp. inner) approximations for C (resp. for its dual), thus complementing
previous inner (resp. outer) approximations for C (for the dual). In
particular, both inner and outer approximations have a very simple
interpretation. Finally, extension to K-copositivity and K-complete positivity
for a closed convex cone K, is straightforward.Comment: 8
Minimal zeros of copositive matrices
Let be an element of the copositive cone . A zero of
is a nonzero nonnegative vector such that . The support of is
the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the
positive entries of . A zero of is called minimal if there does not
exist another zero of such that its support \mbox{supp}v is a strict
subset of \mbox{supp}u. We investigate the properties of minimal zeros of
copositive matrices and their supports. Special attention is devoted to
copositive matrices which are irreducible with respect to the cone of
positive semi-definite matrices, i.e., matrices which cannot be written as a
sum of a copositive and a nonzero positive semi-definite matrix. We give a
necessary and sufficient condition for irreducibility of a matrix with
respect to in terms of its minimal zeros. A similar condition is given
for the irreducibility with respect to the cone of entry-wise
nonnegative matrices. For matrices which are irreducible with respect
to both and are extremal. For a list of candidate
combinations of supports of minimal zeros which an exceptional extremal matrix
can have is provided.Comment: Some conditions and proofs simplifie
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