A factorization method for completely positive matrices

Abstract A matrix A is called completely positive, if there exists an entrywise nonnegative matrix B such that A = B B T . These matrices play a major role in combinatorial and quadratic optimization. In this paper, we study the problem of finding a nonnegative factorization B B T of a given completely positive matrix A. We formulate this factorization problem as a nonconvex feasibility problem and develop a solution method based on alternating projections. A local convergence result can be shown for this algorithm. We also provide a heuristic extension which improves the numerical performance of the algorithm. Extensive numerical tests show that the factorization method is very fast in most of the test instances and performs better than existing algorithms.

[1]  Mathieu Dutour Sikiric,et al.  A simplex algorithm for rational cp-factorization , 2018, Mathematical Programming.

[2]  Florian Jarre,et al.  On the computation of C* certificates , 2009, J. Glob. Optim..

[3]  Adrian S. Lewis,et al.  Alternating Projections on Manifolds , 2008, Math. Oper. Res..

[4]  Etienne de Klerk,et al.  On Copositive Programming and Standard Quadratic Optimization Problems , 2000, J. Glob. Optim..

[5]  Chris H. Q. Ding,et al.  On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering , 2005, SDM.

[6]  D. Szyld,et al.  Nearly positive matrices , 2014 .

[7]  Samuel Burer,et al.  On the copositive representation of binary and continuous nonconvex quadratic programs , 2009, Math. Program..

[8]  D. Bernstein.  Matrix Mathematics: Theory, Facts, and Formulas , 2009 .

[9]  J. Wells,et al.  Embeddings and Extensions in Analysis , 1975 .

[10]  Peter J. C. Dickinson.  AN IMPROVED CHARACTERISATION OF THE INTERIOR OF THE COMPLETELY POSITIVE CONE , 2010 .

[11]  A. Berman,et al.  Completely Positive Matrices , 2003 .

[12]  Jiawang Nie,et al.  The $${\mathcal {A}}$$A-Truncated $$K$$K-Moment Problem , 2012, Found. Comput. Math..

[13]  Mirjam Dür,et al.  Open problems in the theory of completely positive and copositive matrices , 2015 .

[14]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[15]  Mirjam Dür,et al.  Factorization and cutting planes for completely positive matrices by copositive projection , 2014, Math. Program..

[16]  Mirjam Dür,et al.  Linear-Time Complete Positivity Detection and Decomposition of Sparse Matrices , 2012, SIAM J. Matrix Anal. Appl..

[17]  Immanuel M. Bomze,et al.  Copositive optimization - Recent developments and applications , 2012, Eur. J. Oper. Res..

[18]  Mirjam Dür,et al.  Interior points of the completely positive cone. , 2008 .

[19]  Etienne de Klerk,et al.  Approximation of the Stability Number of a Graph via Copositive Programming , 2002, SIAM J. Optim..

[20]  Robert E. Mahony,et al.  Optimization Algorithms on Matrix Manifolds , 2007 .

[21]  Mirjam Dür,et al.  Copositive Programming – a Survey , 2010 .

[22]  Immanuel M. Bomze,et al.  The structure of completely positive matrices according to their CP-rank and CP-plus-rank , 2015 .

[23]  Pablo A. Parrilo,et al.  Semidefinite Descriptions of the Convex Hull of Rotation Matrices , 2014, SIAM J. Optim..

[24]  Immanuel M. Bomze,et al.  Building a completely positive factorization , 2017, Central Eur. J. Oper. Res..

[25]  Changqing Xu.  Completely positive matrices , 2004 .

[26]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[27]  Immanuel M. Bomze,et al.  From seven to eleven: Completely positive matrices with high cp-rank , 2014 .

[28]  John Hannah,et al.  Nonnegative factorization of completely positive matrices , 1983 .

[29]  Peter J. C. Dickinson,et al.  On the computational complexity of membership problems for the completely positive cone and its dual , 2014, Comput. Optim. Appl..

[30]  D. Cooley,et al.  Decompositions of dependence for high-dimensional extremes , 2016, Biometrika.

[31]  Dmitriy Drusvyatskiy,et al.  Transversality and Alternating Projections for Nonconvex Sets , 2014, Found. Comput. Math..

[32]  Changqing Xu,et al.  A simple sufficient condition for complete positivity , 2015 .