Economics at your fingertips  

Algorithms for positive semidefinite factorization

Arnaud Vandaele (), François Glineur and Nicolas Gillis ()
Additional contact information
Arnaud Vandaele: Université de Mons
Nicolas Gillis: Université de Mons

Computational Optimization and Applications, 2018, vol. 71, issue 1, 193-219

Abstract: Abstract 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 $$\{A^1,\ldots ,A^m\}$$ { A 1 , … , A m } and $$\{B^1,\ldots ,B^n\}$$ { B 1 , … , B n } such that $$X_{i,j}=\text {trace}(A^iB^j)$$ X i , j = trace ( A i B j ) for $$i=1,\ldots ,m$$ i = 1 , … , m , and $$j=1,\ldots ,n$$ 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 $$k=1+ \lceil \log _2(n) \rceil $$ k = 1 + ⌈ log 2 ( n ) ⌉ for the regular n-gons when $$n=5$$ 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 $$A^i$$ A i and $$B^j$$ B j have rank one) and completely PSD factorizations (which is the special case where the input matrix is symmetric and equality $$A^i=B^i$$ A i = B i is required for all i).

Keywords: Positive semidefinite factorization; Extended formulations; Fast gradient method; Coordinate descent method (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations: Track citations by RSS feed

Downloads: (external link) Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link:

Ordering information: This journal article can be ordered from

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla ().

Page updated 2019-07-22
Handle: RePEc:spr:coopap:v:71:y:2018:i:1:d:10.1007_s10589-018-9998-x