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Improved SDP bounds for minimizing quadratic functions over the \(\ell^{1}\)-ball

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Abstract

Nesterov suggested an SDP-based bound for the problem to minimize a quadratic form over the \(\ell ^1\)-ball. In this note, we introduce a tighter SDP-based bound, the so-called copositive bound, and illustrate the improvement by simulation results. The copositive bound has the additional advantage that it can be easily extended to the inhomogeneous case of quadratic objectives including a linear term. We also indicate some improvements of the eigenvalue bound for the quadratic optimization over the \(\ell ^p\)-ball with 1 < p < 2, at least for p close to one.

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References

  1. Anstreicher K., Burer S. (2005) D.C. Versus Copositive Bounds for Standard QP. J. Global Optim. 33, 299–312

    Article  MATH  MathSciNet  Google Scholar 

  2. Diananda P.H. (1967) On non-negative forms in real variables some or all of which are non-negative. Proc. Cambridge Philos. Soc. 58, 17–25

    Article  MathSciNet  Google Scholar 

  3. Bomze I.M. (2005) Portfolio selection via replicator dynamics and projections of indefinite estimated covariances. Dyn. Contin. Discret. Impuls. Syst. B 12, 527–564

    MATH  MathSciNet  Google Scholar 

  4. Bomze I.M., Dür M., de Klerk E., Quist A., Roos C., Terlaky T. (2000) On copositive programming and standard quadratic optimization problems. J. Global Optim. 18, 301–320

    Article  MATH  MathSciNet  Google Scholar 

  5. Bomze I.M., de Klerk E. (2002) Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim. 24, 163–185

    Article  MATH  MathSciNet  Google Scholar 

  6. Bomze, I.M., Locatelli, M., Tardella, F.: Efficient and cheap bounds for (Standard) Quadratic Optimization, Technical Report dis tr 2005/10, Dipartimento di Informatica e Sistemistica “Antonio Ruberti”, Universitá degli Studi di Roma “La Sapienza”, also available at www.optimization-online.org/DB_HTML/2005/07/1176.html (2005)

  7. Feingold D.G., Varga R.S. (1962) Block diagonally dominant matrices and generalizations of the gerschgorin circle theorem. Pacific J. Mathe. 12, 1241–1250

    MATH  MathSciNet  Google Scholar 

  8. Nesterov Yu. (2000) Global quadratic optimization via conic relaxation. In: Wolkowicz H., Saigal R., Vandenberghe L. (eds) Handbook of Semidefinite Programming. Kluwer Academic Publishers, Dordrecht, pp. 363–387

    Google Scholar 

  9. Sturm J.F. (1999) Using SeDuMi 1.02, a MATLAB Toolbox for Optimization Over Symmetric Cones. Optim. Methods Softw. 11-12, 625–653

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Authors and Affiliations

  1. ISDS, University of Vienna, Universitätsstraße 5/9, 1010, Wien, Austria

    Immanuel M. Bomze, Florian Frommlet & Martin Rubey

Authors
  1. Immanuel M. Bomze
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  2. Florian Frommlet
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  3. Martin Rubey
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Correspondence to Immanuel M. Bomze.

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Bomze, I.M., Frommlet, F. & Rubey, M. Improved SDP bounds for minimizing quadratic functions over the \(\ell^{1}\)-ball. Optimization Letters 1, 49–59 (2007). https://doi.org/10.1007/s11590-006-0018-1

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  • DOI: https://doi.org/10.1007/s11590-006-0018-1

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