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Building a completely positive factorization.

Immanuel M Bomze¹

¹ISOR and VCOR, University of Vienna, Vienna, Austria.

Central European Journal of Operations Research

|May 19, 2018

Summary

Researchers developed a new method to extend completely positive (cp) factorizations of matrices. This approach, using a bordering technique, shows promise for constructing cp factorizations in higher dimensions.

Area of Science:

Linear Algebra
Matrix Theory
Optimization

Background:

A symmetric matrix is completely positive (cp) if it can be factored into a product of a rectangular matrix and its transpose.
This cp factorization is equivalent to representing the matrix as a Gram matrix of vectors in the positive orthant.
Finding such factorizations involves angle packing and determining embedding dimensions, with potential non-uniqueness.

Purpose of the Study:

To establish sufficient conditions for extending an existing cp factorization of a principal submatrix to the entire matrix.
To develop a novel bordering approach for constructing cp factorizations.
To assess the applicability and potential of this method in higher dimensions.

Main Methods:

Utilizing a bordering approach to extend known cp factorizations.

Keywords:

Copositive optimization Schur complement cp-rank

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Building upon existing cp factorizations of principal blocks within the matrix.

Employing simulations to evaluate the method's performance.

Main Results:

Sufficient conditions were established for the successful extension of cp factorizations.
The bordering approach proved effective in constructing cp factorizations for larger matrices.
Simulations indicated the method's promise, even for matrices of higher dimensions.

Conclusions:

The developed bordering approach provides a viable strategy for extending completely positive factorizations.
This method offers a promising computational tool for problems involving Gram matrices and angle packing.
The technique demonstrates potential for application in complex, high-dimensional scenarios.