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Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming
1. | Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, Netherlands |
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. |
[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. |
[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. |
[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. |
[5] |
P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, E. Udovina and D. Vaintrob, Introduction to representation theory, preprint,, , ().
|
[6] |
D. Gijswijt, "Matrix Algebras and Semidefinite Programming Techniques for Codes," Ph. D. Thesis, University of Amsterdam, The Netherlands, 2005. |
[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. |
[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. |
[9] |
C. Godsil, "Association Schemes," Lecture notes, University of Waterloo, 2010. Available from: http://quoll.uwaterloo.ca/mine/Notes/assoc2.pdf. |
[10] |
A. Graham, "Kroneker Products and Matrix Calculus with Applications," John Wiley and Sons, Chichester, 1981.
doi: ISBN-13/978-0-4702-7300-5. |
[11] |
D. G. Higman, Coherent algebras, Linear Algebra Applications, 93 (1987), 209-239.
doi: 10.1016/S0024-3795(87)90326-0. |
[12] |
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, 1990.
doi: ISBN-13/978-0-5213-8632-6. |
[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. |
[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. |
[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. |
[16] |
E. de Klerk, C. Dobre, D. V. Pasechnik and R. Sotirov, On semidefinite programming relaxations of maximum k-section,, Mathematical Programming-B, (): 10107.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
M. Laurent, Strengthened semidefinite bounds for codes, Mathematical Programming, 109 (2007), 239-261.
doi: 10.1007/s10107-006-0030-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
F. Vallentin, Symmetry in semidefinite programs, Linear Algebra and Applications, 430 (2009), 360-369.
doi: 10.1016/j.laa.2008.07.025. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, E. Udovina and D. Vaintrob, Introduction to representation theory, preprint,, , ().
|
[6] |
D. Gijswijt, "Matrix Algebras and Semidefinite Programming Techniques for Codes," Ph. D. Thesis, University of Amsterdam, The Netherlands, 2005. |
[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. |
[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. |
[9] |
C. Godsil, "Association Schemes," Lecture notes, University of Waterloo, 2010. Available from: http://quoll.uwaterloo.ca/mine/Notes/assoc2.pdf. |
[10] |
A. Graham, "Kroneker Products and Matrix Calculus with Applications," John Wiley and Sons, Chichester, 1981.
doi: ISBN-13/978-0-4702-7300-5. |
[11] |
D. G. Higman, Coherent algebras, Linear Algebra Applications, 93 (1987), 209-239.
doi: 10.1016/S0024-3795(87)90326-0. |
[12] |
R. A. Horn and C. R. Johnson, "Matrix Analysis," Cambridge University Press, 1990.
doi: ISBN-13/978-0-5213-8632-6. |
[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. |
[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. |
[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. |
[16] |
E. de Klerk, C. Dobre, D. V. Pasechnik and R. Sotirov, On semidefinite programming relaxations of maximum k-section,, Mathematical Programming-B, (): 10107.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
M. Laurent, Strengthened semidefinite bounds for codes, Mathematical Programming, 109 (2007), 239-261.
doi: 10.1007/s10107-006-0030-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[30] |
F. Vallentin, Symmetry in semidefinite programs, Linear Algebra and Applications, 430 (2009), 360-369.
doi: 10.1016/j.laa.2008.07.025. |
[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. |
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