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Copositive Programming via Semi-Infinite Optimization

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Abstract

Copositive programming (CP) can be regarded as a special instance of linear semi-infinite programming (SIP). We study CP from the viewpoint of SIP and discuss optimality and duality results. Different approximation schemes for solving CP are interpreted as discretization schemes in SIP. This leads to sharp explicit error bounds for the values and solutions in dependence on the mesh size. Examples illustrate the structure of the original program and the approximation schemes.

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References

  1. Bomze, I.M., De Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Glob. Optim. 24(2), 163–185 (2002)

    Article  MATH  Google Scholar 

  2. de Klerk, E., Pasechnik, D.V.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Alper Yıldırım, E.: On the accuracy of uniform polyhedral approximations of the copositive cone. Optim. Methods Softw. 27(1), 155–173 (2012)

    Article  MathSciNet  Google Scholar 

  5. Bundfuss, S., Dür, M.: An adaptive linear approximation algorithm for copositive programs. SIAM J. Optim. 20(1), 30–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berman, A., Shaked-Monderer, N.: Completely Positive Matrices. World Scientific, Singapore (2003)

    Book  MATH  Google Scholar 

  7. Dickinson, P.J.: An improved characterisation of the interior of the completely positive cone. Electron. J. Linear Algebra 20, 723–729 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Dür, M., Still, G.: Interior points of the completely positive cone. Electron. J. Linear Algebra 17, 48–53 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  10. López, M., Still, G.: Semi-infinite programming (invited review). Eur. J. Oper. Res. 180(2), 491–518 (2007)

    Article  MATH  Google Scholar 

  11. Faigle, U., Kern, W., Still, G.: Algorithmic Principles of Mathematical Programming. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  12. Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35(3), 380–429 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bomze, I.M., Schachinger, W., Uchida, G.: Think co (mpletely) positive! Matrix properties, examples and a clustered bibliography on copositive optimization. J. Glob. Optim. 52(3), 423–445 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Still, G.: Discretization in semi-infinite programming: the rate of convergence. Math. Program. 91(1), 53–69 (2001)

    MathSciNet  MATH  Google Scholar 

  15. Eichfelder, G., Jahn, J.: Set-semidefinite optimization. J. Convex Anal. 15(4), 767–801 (2008)

    MathSciNet  MATH  Google Scholar 

  16. Burer, S., Dong, H.: Representing quadratically constrained quadratic programs as generalized copositive programs. Oper. Res. Lett. 40(3), 203–206 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous referees for many valuable comments and suggestions.

This work was done while Mirjam Dür was working at Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, The Netherlands. She was supported by the Netherlands Organization for Scientific Research (NWO) through Vici Grant No. 639.033.907.

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Authors and Affiliations

  1. Department of Mathematics, University of Twente, 7500 AE, Enschede, the Netherlands

    Faizan Ahmed & Georg Still

  2. Department of Mathematics, University of Trier, 54286, Trier, Germany

    Mirjam Dür

Authors
  1. Faizan Ahmed
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  2. Mirjam Dür
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  3. Georg Still
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Correspondence to Georg Still.

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Communicated by Johannes Jahn.

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Ahmed, F., Dür, M. & Still, G. Copositive Programming via Semi-Infinite Optimization. J Optim Theory Appl 159, 322–340 (2013). https://doi.org/10.1007/s10957-013-0344-2

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  • DOI: https://doi.org/10.1007/s10957-013-0344-2

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