A semidefinite relaxation algorithm for checking completely positive separable matrices

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April 2019, 15(2): 893-908. doi: 10.3934/jimo.2018076

A semidefinite relaxation algorithm for checking completely positive separable matrices

Anwa Zhou ^1,, and Jinyan Fan ^2,

1.	Department of Mathematics, Shanghai University, Shanghai 200444, China
2.	School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

* Corresponding author: Anwa Zhou

Received July 2017 Revised November 2017 Published April 2019 Early access June 2018

A symmetric matrix $A$ is completely positive (CP) if there exists an entrywise nonnegative matrix $V$ such that $A = VV ^T$. A real symmetric matrix is called completely positive separable (CPS) if it can be written as a sum of rank-1 Kronecker products of completely positive matrices. This paper studies the CPS problem. A criterion is given to determine whether a given matrix is CPS, and a specific CPS decomposition is constructed if the matrix is CPS.

Keywords: Completely positive matrices, completely positive separability, moment problems, semidefinite algorithm.

Mathematics Subject Classification: Primary: 15A23, 15B48, 44A60, 90C22.

Citation: Anwa Zhou, Jinyan Fan. A semidefinite relaxation algorithm for checking completely positive separable matrices. Journal of Industrial and Management Optimization, 2019, 15 (2) : 893-908. doi: 10.3934/jimo.2018076

References:

[1]	N. Arima, S. Kim and M. Kojima, A quadratically constrained quadratic optimization model for completely positive cone programming, SIAM J. Optim., 23 (2013), 2320-2340. doi: 10.1137/120890636.
[2]	A. Berman and N. Shaked-Monderer, Completely Positive Matrices, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795212.
[3]	S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Math. Program., Ser. A, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.
[4]	R. Curto and L. Fialkow, Truncated K-moment problems in several variables, J. Operator Theory, 54 (2005), 189-226.
[5]	G. Dahl, J. M. Leinaas, J. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725. doi: 10.1016/j.laa.2006.08.026.
[6]	J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9781611971446.
[7]	A. C. Doherty, P. A. Parrilo and F. M. Spedalieri, Complete family of separability criteria, Phy. Rev. A, 69 (2004), 022308. doi: 10.1103/PhysRevA.69.022308.
[8]	P. J. Dickinson and L. Gijben, On the computational complexity of membership problems for the completely positive cone and its dual, Comput. Optim. Appl., 57 (2014), 403-415. doi: 10.1007/s10589-013-9594-z.
[9]	I. Dukanovic and F. Rendl, Copositive programming motivated bounds on the stability and the chromatic numbers, Math. Program., Ser. A, 121 (2010), 249-268. doi: 10.1007/s10107-008-0233-x.
[10]	L. Gurvits, Classical deterministic complexity of Edmonds' problem and quantum entanglement, in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, ACM, New York, (2003), 10-19. doi: 10.1145/780542.780545.
[11]	L. Gurvits and H. Barnum, Largest separable balls around the maximally mixed bipartite quantum state, Phys. Rev. A, 66 (2002), 062311. doi: 10.1103/PhysRevA.66.062311.
[12]	D. Henrion and J. B. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly, Positive polynomials in control, Lecture Notes in Control and Inform. Sci. , Springer, Berlin, 312 (2005), 293-310. doi: 10.1007/10997703_15.
[13]	D. Henrion, J. B. Lasserre and J. Loefberg, GloptiPoly 3: Moments, optimization and semidefinite programming, Optim. Methods Softw., 24 (2009), 761-779. doi: 10.1080/10556780802699201.
[14]	V. Jeyakumar, J. B. Lasserre and G. Li, On Polynomial Optimization over Non-compact Semi-algebraic Sets, J. Optim. Theory Appl., 163 (2014), 707-718. doi: 10.1007/s10957-014-0545-3.
[15]	J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), 796-817. doi: 10.1137/S1052623400366802.
[16]	J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, London, 2010.
[17]	M. Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging Applications of Algebraic Geometry, Springer, New York, 149 (2009), 157-270. doi: 10.1007/978-0-387-09686-5_7.
[18]	Z. Luo and L. Qi, Completely positive tensors: Properties, easily checkable subclasses, and tractable relaxations, SIAM. J. Matrix Anal. Appl., 37 (2016), 1675-1698. doi: 10.1137/15M1025220.
[19]	K. G. Murty and S. N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Math. Program., 39 (1987), 117-129. doi: 10.1007/BF02592948.
[20]	J. Nie, Certifying convergence of Lasserre's hierarchy via flat truncation, Math. Program., Ser. A, 142 (2013), 485-510. doi: 10.1007/s10107-012-0589-9.
[21]	J. Nie, The $\mathcal{A}$-truncated K-moment problem, Found. Comput. Math., 14 (2014), 1243-1276. doi: 10.1007/s10208-014-9225-9.
[22]	J. Nie, Linear optimization with cones of moments and nonnegative polynomials, Math. Program., Ser. B, 153 (2015), 247-274. doi: 10.1007/s10107-014-0797-6.
[23]	J. Nie, Optimality conditions and finite convergence of Lasserre's hierarchy, Math. Program., Ser. A, 146 (2014), 97-121. doi: 10.1007/s10107-013-0680-x.
[24]	J. Nie and X. Zhang, Positive maps and separable matrices, SIAM J. Optim., 26 (2016), 1236-1256. doi: 10.1137/15M1018514.
[25]	M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., 42 (1993), 969-984. doi: 10.1512/iumj.1993.42.42045.
[26]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.
[27]	L. Qi, F. Wang and Y. Wang, Z-Eigenvalue methods for a global optimization polynomial optimization problem, Math. Program., Ser. A, 118 (2009), 301-306. doi: 10.1007/s10107-007-0193-6.
[28]	P. Rosakis, Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics, Arch. Rational Mech. Anal., 109 (1990), 1-37. doi: 10.1007/BF00377977.
[29]	H. C. Simpson and S. J. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Rational Mech. Anal., 84 (1983), 55-68. doi: 10.1007/BF00251549.
[30]	J. F. Sturm, SeDuMi 1.02: A MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11/12 (1999), 625-653. doi: 10.1080/10556789908805766.
[31]	M. J. Todd and Y. Ye, Approximate Farkas lemmas and stopping rules for iterative infeasible-point algorithms for linear programming, Math. Program., Ser. A, 81 (1998), 1-21. doi: 10.1007/BF01584841.
[32]	Y. Wang and M. Aron, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J. Elasticity, 44 (1996), 89-96. doi: 10.1007/BF00042193.
[33]	A. Zhou and J. Fan, The CP-matrix completion problem, SIAM. J. Matrix Anal. Appl., 35 (2014), 127-142. doi: 10.1137/130919490.

show all references

References:

[1]	N. Arima, S. Kim and M. Kojima, A quadratically constrained quadratic optimization model for completely positive cone programming, SIAM J. Optim., 23 (2013), 2320-2340. doi: 10.1137/120890636.
[2]	A. Berman and N. Shaked-Monderer, Completely Positive Matrices, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. doi: 10.1142/9789812795212.
[3]	S. Burer, On the copositive representation of binary and continuous nonconvex quadratic programs, Math. Program., Ser. A, 120 (2009), 479-495. doi: 10.1007/s10107-008-0223-z.
[4]	R. Curto and L. Fialkow, Truncated K-moment problems in several variables, J. Operator Theory, 54 (2005), 189-226.
[5]	G. Dahl, J. M. Leinaas, J. Myrheim and E. Ovrum, A tensor product matrix approximation problem in quantum physics, Linear Algebra Appl., 420 (2007), 711-725. doi: 10.1016/j.laa.2006.08.026.
[6]	J. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, PA, 1997. doi: 10.1137/1.9781611971446.
[7]	A. C. Doherty, P. A. Parrilo and F. M. Spedalieri, Complete family of separability criteria, Phy. Rev. A, 69 (2004), 022308. doi: 10.1103/PhysRevA.69.022308.
[8]	P. J. Dickinson and L. Gijben, On the computational complexity of membership problems for the completely positive cone and its dual, Comput. Optim. Appl., 57 (2014), 403-415. doi: 10.1007/s10589-013-9594-z.
[9]	I. Dukanovic and F. Rendl, Copositive programming motivated bounds on the stability and the chromatic numbers, Math. Program., Ser. A, 121 (2010), 249-268. doi: 10.1007/s10107-008-0233-x.
[10]	L. Gurvits, Classical deterministic complexity of Edmonds' problem and quantum entanglement, in Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, ACM, New York, (2003), 10-19. doi: 10.1145/780542.780545.
[11]	L. Gurvits and H. Barnum, Largest separable balls around the maximally mixed bipartite quantum state, Phys. Rev. A, 66 (2002), 062311. doi: 10.1103/PhysRevA.66.062311.
[12]	D. Henrion and J. B. Lasserre, Detecting global optimality and extracting solutions in GloptiPoly, Positive polynomials in control, Lecture Notes in Control and Inform. Sci. , Springer, Berlin, 312 (2005), 293-310. doi: 10.1007/10997703_15.
[13]	D. Henrion, J. B. Lasserre and J. Loefberg, GloptiPoly 3: Moments, optimization and semidefinite programming, Optim. Methods Softw., 24 (2009), 761-779. doi: 10.1080/10556780802699201.
[14]	V. Jeyakumar, J. B. Lasserre and G. Li, On Polynomial Optimization over Non-compact Semi-algebraic Sets, J. Optim. Theory Appl., 163 (2014), 707-718. doi: 10.1007/s10957-014-0545-3.
[15]	J. B. Lasserre, Global optimization with polynomials and the problem of moments, SIAM J. Optim., 11 (2001), 796-817. doi: 10.1137/S1052623400366802.
[16]	J. B. Lasserre, Moments, Positive Polynomials and Their Applications, Imperial College Press, London, 2010.
[17]	M. Laurent, Sums of squares, moment matrices and optimization over polynomials, Emerging Applications of Algebraic Geometry, Springer, New York, 149 (2009), 157-270. doi: 10.1007/978-0-387-09686-5_7.
[18]	Z. Luo and L. Qi, Completely positive tensors: Properties, easily checkable subclasses, and tractable relaxations, SIAM. J. Matrix Anal. Appl., 37 (2016), 1675-1698. doi: 10.1137/15M1025220.
[19]	K. G. Murty and S. N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Math. Program., 39 (1987), 117-129. doi: 10.1007/BF02592948.
[20]	J. Nie, Certifying convergence of Lasserre's hierarchy via flat truncation, Math. Program., Ser. A, 142 (2013), 485-510. doi: 10.1007/s10107-012-0589-9.
[21]	J. Nie, The $\mathcal{A}$-truncated K-moment problem, Found. Comput. Math., 14 (2014), 1243-1276. doi: 10.1007/s10208-014-9225-9.
[22]	J. Nie, Linear optimization with cones of moments and nonnegative polynomials, Math. Program., Ser. B, 153 (2015), 247-274. doi: 10.1007/s10107-014-0797-6.
[23]	J. Nie, Optimality conditions and finite convergence of Lasserre's hierarchy, Math. Program., Ser. A, 146 (2014), 97-121. doi: 10.1007/s10107-013-0680-x.
[24]	J. Nie and X. Zhang, Positive maps and separable matrices, SIAM J. Optim., 26 (2016), 1236-1256. doi: 10.1137/15M1018514.
[25]	M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana Univ. Math. J., 42 (1993), 969-984. doi: 10.1512/iumj.1993.42.42045.
[26]	L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), 1302-1324. doi: 10.1016/j.jsc.2005.05.007.
[27]	L. Qi, F. Wang and Y. Wang, Z-Eigenvalue methods for a global optimization polynomial optimization problem, Math. Program., Ser. A, 118 (2009), 301-306. doi: 10.1007/s10107-007-0193-6.
[28]	P. Rosakis, Ellipticity and deformations with discontinuous deformation gradients in finite elastostatics, Arch. Rational Mech. Anal., 109 (1990), 1-37. doi: 10.1007/BF00377977.
[29]	H. C. Simpson and S. J. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials, Arch. Rational Mech. Anal., 84 (1983), 55-68. doi: 10.1007/BF00251549.
[30]	J. F. Sturm, SeDuMi 1.02: A MATLAB toolbox for optimization over symmetric cones, Optim. Methods Softw., 11/12 (1999), 625-653. doi: 10.1080/10556789908805766.
[31]	M. J. Todd and Y. Ye, Approximate Farkas lemmas and stopping rules for iterative infeasible-point algorithms for linear programming, Math. Program., Ser. A, 81 (1998), 1-21. doi: 10.1007/BF01584841.
[32]	Y. Wang and M. Aron, A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media, J. Elasticity, 44 (1996), 89-96. doi: 10.1007/BF00042193.
[33]	A. Zhou and J. Fan, The CP-matrix completion problem, SIAM. J. Matrix Anal. Appl., 35 (2014), 127-142. doi: 10.1137/130919490.

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