American Institute of Mathematical Sciences

Advanced
2013, 3(2): 367-378. doi: 10.3934/naco.2013.3.367

Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming

Cristian Dobre 1,

1. 

Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, Netherlands

Received  June 2011 Revised  November 2012 Published  April 2013

In this paper we give a proof for the special structure of the Wedderburn decomposition of the regular *-representation of a given matrix $*$-algebra. This result was stated without proof in: de Klerk, E., Dobre, C. and Pasechnik, D.V.: Numerical block diagonalization of matrix $*$-algebras with application to semidefinite programming, Mathematical Programming-B, 129 (2011), 91--111; and is used in applications of semidefinite programming (SDP) for structured combinatorial optimization problems. In order to provide the proof for this special structure we derive several other mathematical properties of the regular *-representation.
Keywords: regular *-representation, algebraic symmetry, pre-processing, semidefinite programming, Matrix algebras.
Mathematics Subject Classification: Primary: 90C22; Secondary: 06B1.
Citation: Cristian Dobre. Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 367-378. doi: 10.3934/naco.2013.3.367
References:
[1]

C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc., 21 (2008), 909-924. doi: 10.1090/S0894-0347-07-00589-9.  Google Scholar

[2]

C. Bachoc, D. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs, in "Handbook on Semidefinite, Conic and Polynomial Optimization" (eds. M. F. Anjos and J. B. Lasserre), Springer, (2012), 219-270. doi: 10.1007/978-1-4614-0769-0_9.  Google Scholar

[3]

Y.-Q. Bai, E. de Klerk, D. V. Pasechnik and R. Sotirov, Exploiting group symmetry in truss topology optimization, Optimization and Engineering, 10 (2009), 331-349. doi: 10.1007/s11081-008-9050-6.  Google Scholar

[4]

P. J. Cameron, Coherent configurations, association schemes and permutation groups, in "Groups, Combinatorics and Geometry" (eds. A.A. Ivanov, M.W. Liebeck and J. Saxl), World Scientific, Singapore, (2003), 55-71.  Google Scholar

[5]

P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, E. Udovina and D. Vaintrob, Introduction to representation theory, preprint,, , ().   Google Scholar

[6]

D. Gijswijt, "Matrix Algebras and Semidefinite Programming Techniques for Codes," Ph. D. Thesis, University of Amsterdam, The Netherlands, 2005. Google Scholar

[7]

D. Gijswijt, A. Schrijver and H. Tanaka, New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming, Journal of Combinatorial Theory, 113 (2006), 1719-1731. doi: 10.1016/j.jcta.2006.03.010.  Google Scholar

[8]

K. Gatermann and P. A. Parrilo, Symmetry groups, semidefinite programs, and sums of squares, J. Pure and Applied Algebra, 192 (2004), 95-128. doi: 10.1016/j.jpaa.2003.12.011.  Google Scholar

[9]

C. Godsil, "Association Schemes," Lecture notes, University of Waterloo, 2010. Available from: http://quoll.uwaterloo.ca/mine/Notes/assoc2.pdf. Google Scholar

[10]

A. Graham, "Kroneker Products and Matrix Calculus with Applications," John Wiley and Sons, Chichester, 1981. doi: ISBN-13/978-0-4702-7300-5.  Google Scholar

[11]

D. G. Higman, Coherent algebras, Linear Algebra Applications, 93 (1987), 209-239. doi: 10.1016/S0024-3795(87)90326-0.  Google Scholar

[12]

R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, 1990. doi: ISBN-13/978-0-5213-8632-6.  Google Scholar

[13]

Y. Kanno, M. Ohsaki, K. Murota and N. Katoh, Group symmetry in interior-point methods for semidefinite program, Optimization and Engineering, 2 (2001), 293-320. doi: 10.1023/A:1015366416311.  Google Scholar

[14]

E. de Klerk, Exploiting special structure in semidefinite programming: a survey, of theory and applications, European Journal of Operational Research, 201 (2010), 1-10. doi: 10.1016/j.ejor.2009.01.025.  Google Scholar

[15]

E. de Klerk, C. Dobre and D. V. Pasechnik, Numerical block diagonalization of matrix *-algebras with application to semidefinite programming, Mathematical Programming-B, 129 (2011), 91-111. doi: 10.1007/s10107-011-0461-3.  Google Scholar

[16]

E. de Klerk, C. Dobre, D. V. Pasechnik and R. Sotirov, On semidefinite programming relaxations of maximum k-section,, Mathematical Programming-B, (): 10107.   Google Scholar

[17]

E. de Klerk and C. Dobre, A comparison of lower bounds for the Symmetric Circulant Traveling Salseman Problem, Discrete Applied Mathematics, 159 (2011), 1815-1826. doi: 10.1016/j.dam.2011.01.026.  Google Scholar

[18]

E. de Klerk, D. V. Pasechnik and A. Schrijver, Reduction of symmetric semidefinite programs using the regular *-representation, Mathematical Programming-B, 109 (2007), 613-624. doi: 10.1007/s10107-006-0039-7.  Google Scholar

[19]

E. de Klerk, M. W. Newman, D. V. Pasechnik and R. Sotirov, On the Lovász $\vartheta$-number of almost regular graphs with application to Erdös-Rényi graphs, European Journal of Combinatorics, 31 (2009), 879-888. doi: 10.1016/j.ejc.2008.07.022.  Google Scholar

[20]

E. de Klerk, J. Maharry, D. V. Pasechnik, B. Richter and G. Salazar, Improved bounds for the crossing numbers of km,n and kn, SIAM Journal on Discrete Mathematics, 20 (2006), 189-202. doi: 10.1137/S0895480104442741.  Google Scholar

[21]

E. de Klerk and R. Sotirov, Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem, Mathematical Programming, 122 (2010), 225-246. doi: 10.1007/s10107-008-0246-5.  Google Scholar

[22]

M. Kojima, S. Kojima and S. Hara, Linear algebra for semidefinite programming, in "Research Report B-290," Tokyo Institute of Technology, (1997), 1-23.  Google Scholar

[23]

M. Laurent, Strengthened semidefinite bounds for codes, Mathematical Programming, 109 (2007), 239-261. doi: 10.1007/s10107-006-0030-3.  Google Scholar

[24]

L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information theory, 25 (1979), 1-7. doi: 10.1109/TIT.1979.1055985.  Google Scholar

[25]

T. Maehara and K. Murota, A numerical algorithm for block-diagonal decomposition of matrix *-algebras with general irreducible components, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 263-293. doi: 10.1007/s13160-010-0007-8.  Google Scholar

[26]

R. J. McEliece, E. R. Rodemich and H. C. Rumsey, The Lovász bound and some generalizations, Journal of Combinatorics, Information & System Sciences, 3 (1978), 134-152.  Google Scholar

[27]

K. Murota, Y. Kanno, M. Kojima and S. Kojima, A numerical algorithm for block-diagonal decomposition of matrix *-algebras with application to semidefinite programming, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 125-160. doi: 10.1007/s13160-010-0006-9.  Google Scholar

[28]

A. Schrijver, A comparison of the Delsarte and Lovász bounds, IEEE Transactions on Information Theory, 25 (1979), 425-429. doi: 10.1109/TIT.1979.1056072.  Google Scholar

[29]

A. Schrijver, New code upper bounds from the Terwilliger algebra, IEEE Transactions on Information Theory, 51 (2005), 2859-2866. doi: 10.1109/TIT.2005.851748.  Google Scholar

[30]

F. Vallentin, Symmetry in semidefinite programs, Linear Algebra and Applications, 430 (2009), 360-369. doi: 10.1016/j.laa.2008.07.025.  Google Scholar

[31]

J. H. M. Wedderburn, On hypercomplex numbers, Proceedings of the London Mathematical Society, 6 (1907), 77-118. doi: 10.1112/plms/s2-6.1.77.  Google Scholar

show all references

References:
[1]

C. Bachoc and F. Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc., 21 (2008), 909-924. doi: 10.1090/S0894-0347-07-00589-9.  Google Scholar

[2]

C. Bachoc, D. Gijswijt, A. Schrijver and F. Vallentin, Invariant semidefinite programs, in "Handbook on Semidefinite, Conic and Polynomial Optimization" (eds. M. F. Anjos and J. B. Lasserre), Springer, (2012), 219-270. doi: 10.1007/978-1-4614-0769-0_9.  Google Scholar

[3]

Y.-Q. Bai, E. de Klerk, D. V. Pasechnik and R. Sotirov, Exploiting group symmetry in truss topology optimization, Optimization and Engineering, 10 (2009), 331-349. doi: 10.1007/s11081-008-9050-6.  Google Scholar

[4]

P. J. Cameron, Coherent configurations, association schemes and permutation groups, in "Groups, Combinatorics and Geometry" (eds. A.A. Ivanov, M.W. Liebeck and J. Saxl), World Scientific, Singapore, (2003), 55-71.  Google Scholar

[5]

P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, E. Udovina and D. Vaintrob, Introduction to representation theory, preprint,, , ().   Google Scholar

[6]

D. Gijswijt, "Matrix Algebras and Semidefinite Programming Techniques for Codes," Ph. D. Thesis, University of Amsterdam, The Netherlands, 2005. Google Scholar

[7]

D. Gijswijt, A. Schrijver and H. Tanaka, New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming, Journal of Combinatorial Theory, 113 (2006), 1719-1731. doi: 10.1016/j.jcta.2006.03.010.  Google Scholar

[8]

K. Gatermann and P. A. Parrilo, Symmetry groups, semidefinite programs, and sums of squares, J. Pure and Applied Algebra, 192 (2004), 95-128. doi: 10.1016/j.jpaa.2003.12.011.  Google Scholar

[9]

C. Godsil, "Association Schemes," Lecture notes, University of Waterloo, 2010. Available from: http://quoll.uwaterloo.ca/mine/Notes/assoc2.pdf. Google Scholar

[10]

A. Graham, "Kroneker Products and Matrix Calculus with Applications," John Wiley and Sons, Chichester, 1981. doi: ISBN-13/978-0-4702-7300-5.  Google Scholar

[11]

D. G. Higman, Coherent algebras, Linear Algebra Applications, 93 (1987), 209-239. doi: 10.1016/S0024-3795(87)90326-0.  Google Scholar

[12]

R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, 1990. doi: ISBN-13/978-0-5213-8632-6.  Google Scholar

[13]

Y. Kanno, M. Ohsaki, K. Murota and N. Katoh, Group symmetry in interior-point methods for semidefinite program, Optimization and Engineering, 2 (2001), 293-320. doi: 10.1023/A:1015366416311.  Google Scholar

[14]

E. de Klerk, Exploiting special structure in semidefinite programming: a survey, of theory and applications, European Journal of Operational Research, 201 (2010), 1-10. doi: 10.1016/j.ejor.2009.01.025.  Google Scholar

[15]

E. de Klerk, C. Dobre and D. V. Pasechnik, Numerical block diagonalization of matrix *-algebras with application to semidefinite programming, Mathematical Programming-B, 129 (2011), 91-111. doi: 10.1007/s10107-011-0461-3.  Google Scholar

[16]

E. de Klerk, C. Dobre, D. V. Pasechnik and R. Sotirov, On semidefinite programming relaxations of maximum k-section,, Mathematical Programming-B, (): 10107.   Google Scholar

[17]

E. de Klerk and C. Dobre, A comparison of lower bounds for the Symmetric Circulant Traveling Salseman Problem, Discrete Applied Mathematics, 159 (2011), 1815-1826. doi: 10.1016/j.dam.2011.01.026.  Google Scholar

[18]

E. de Klerk, D. V. Pasechnik and A. Schrijver, Reduction of symmetric semidefinite programs using the regular *-representation, Mathematical Programming-B, 109 (2007), 613-624. doi: 10.1007/s10107-006-0039-7.  Google Scholar

[19]

E. de Klerk, M. W. Newman, D. V. Pasechnik and R. Sotirov, On the Lovász $\vartheta$-number of almost regular graphs with application to Erdös-Rényi graphs, European Journal of Combinatorics, 31 (2009), 879-888. doi: 10.1016/j.ejc.2008.07.022.  Google Scholar

[20]

E. de Klerk, J. Maharry, D. V. Pasechnik, B. Richter and G. Salazar, Improved bounds for the crossing numbers of km,n and kn, SIAM Journal on Discrete Mathematics, 20 (2006), 189-202. doi: 10.1137/S0895480104442741.  Google Scholar

[21]

E. de Klerk and R. Sotirov, Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem, Mathematical Programming, 122 (2010), 225-246. doi: 10.1007/s10107-008-0246-5.  Google Scholar

[22]

M. Kojima, S. Kojima and S. Hara, Linear algebra for semidefinite programming, in "Research Report B-290," Tokyo Institute of Technology, (1997), 1-23.  Google Scholar

[23]

M. Laurent, Strengthened semidefinite bounds for codes, Mathematical Programming, 109 (2007), 239-261. doi: 10.1007/s10107-006-0030-3.  Google Scholar

[24]

L. Lovász, On the Shannon capacity of a graph, IEEE Transactions on Information theory, 25 (1979), 1-7. doi: 10.1109/TIT.1979.1055985.  Google Scholar

[25]

T. Maehara and K. Murota, A numerical algorithm for block-diagonal decomposition of matrix *-algebras with general irreducible components, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 263-293. doi: 10.1007/s13160-010-0007-8.  Google Scholar

[26]

R. J. McEliece, E. R. Rodemich and H. C. Rumsey, The Lovász bound and some generalizations, Journal of Combinatorics, Information & System Sciences, 3 (1978), 134-152.  Google Scholar

[27]

K. Murota, Y. Kanno, M. Kojima and S. Kojima, A numerical algorithm for block-diagonal decomposition of matrix *-algebras with application to semidefinite programming, Japan Journal of Industrial and Applied Mathematics, 27 (2010), 125-160. doi: 10.1007/s13160-010-0006-9.  Google Scholar

[28]

A. Schrijver, A comparison of the Delsarte and Lovász bounds, IEEE Transactions on Information Theory, 25 (1979), 425-429. doi: 10.1109/TIT.1979.1056072.  Google Scholar

[29]

A. Schrijver, New code upper bounds from the Terwilliger algebra, IEEE Transactions on Information Theory, 51 (2005), 2859-2866. doi: 10.1109/TIT.2005.851748.  Google Scholar

[30]

F. Vallentin, Symmetry in semidefinite programs, Linear Algebra and Applications, 430 (2009), 360-369. doi: 10.1016/j.laa.2008.07.025.  Google Scholar

[31]

J. H. M. Wedderburn, On hypercomplex numbers, Proceedings of the London Mathematical Society, 6 (1907), 77-118. doi: 10.1112/plms/s2-6.1.77.  Google Scholar

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