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A note on sparse SOS and SDP relaxations for polynomial optimization problems over symmetric cones

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Abstract

This short note extends the sparse SOS (sum of squares) and SDP (semidefinite programming) relaxation proposed by Waki, Kim, Kojima and Muramatsu for normal POPs (polynomial optimization problems) to POPs over symmetric cones, and establishes its theoretical convergence based on the recent convergence result by Lasserre on the sparse SOS and SDP relaxation for normal POPs. A numerical example is also given to exhibit its high potential.

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References

  1. Blair, J.R.S., Peyton, B.: An introduction to chordal graphs and clique trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation, pp. 1–29. Springer, New York (1993)

    Google Scholar 

  2. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford University Press, New York (1994)

    MATH  Google Scholar 

  3. Henrion, D., Lasserre, J.B.: Convergent relaxations of polynomial matrix inequalities and static output feedback. IEEE Trans. Autom. Contr. 51, 192–202 (2006)

    Article  MathSciNet  Google Scholar 

  4. Hol, C.W.J., Scherer, C.W.: Sum of squares relaxations for polynomial semidefinite programming, In: Proc. Symp. on Mathematical Theory of Networks and Systems (MTNS), Leuven, Belgium (2004)

  5. Kim, S., Kojima, M., Waki, H.: Generalized Lagrangian duals and sums of squares relaxations of sparse polynomial optimization problems. SIAM J. Optim. 15, 697–719 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Kojima, M., Kim, S., Waki, H.: Sparsity in sums of squares of polynomials. Math. Program. 103, 45–62 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kojima, M.: Sums of squares relaxations of polynomial semidefinite programs. B-397, Dept. of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, November 2003

  8. Kojima, M., Muramatsu, M.: An extension of sums of squares relaxations to polynomial optimization problems over symmetric cones. Math. Program. 110, 315–336 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lasserre, J.B.: Global optimization with polynomials and the problems of moments. SIAM J. Optim. 11, 796–817 (2007)

    Article  MathSciNet  Google Scholar 

  10. Lasserre, J.B.: Convergent SDP-relaxation in polynomial optimization with sparsity. SIAM J. Optim. 17, 822–843 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96, 293–320 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana University Math. J. 42, 969–984 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  13. Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17, 218–242 (2006)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguro-ku, Tokyo, 152-8552, Japan

    Masakazu Kojima

  2. Department of Computer Science, The University of Electro-Communications, Chofugaoka, Chofu-Shi, Tokyo, 182-8585, Japan

    Masakazu Muramatsu

Authors
  1. Masakazu Kojima
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  2. Masakazu Muramatsu
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Corresponding author

Correspondence to Masakazu Kojima.

Additional information

Research of M. Kojima supported by Grant-in-Aid for Scientific Research on Priority Areas 16016234.

Research of M. Muramatsu supported in part by Grant-in-Aid for Young Scientists (B) 15740054.

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Kojima, M., Muramatsu, M. A note on sparse SOS and SDP relaxations for polynomial optimization problems over symmetric cones. Comput Optim Appl 42, 31–41 (2009). https://doi.org/10.1007/s10589-007-9112-2

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  • DOI: https://doi.org/10.1007/s10589-007-9112-2

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