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Decomposition Methods for Solving Nonconvex Quadratic Programs via Branch and Bound*

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Abstract

The aim of this paper is to suggest branch and bound schemes, based on a relaxation of the objective function, to solve nonconvex quadratic programs over a compact polyhedral feasible region. The various schemes are based on different d.c. decomposition methods applied to the quadratic objective function. To improve the tightness of the relaxations, we also suggest solving the relaxed problems with an algorithm based on the so called “optimal level solutions” parametrical approach.

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References

  1. O. Barrientos R. Correa (2000) ArticleTitleAn algorithm for global minimization of linearly constrained quadratic functions Journal of Global Optimization 16 77–93 Occurrence Handle10.1023/A:1008306625093

    Article  Google Scholar 

  2. M.J. Best B. Ding (1997) ArticleTitleGlobal and local quadratic minimization” Journal of Global Optimization 10 77–90 Occurrence Handle10.1023/A:1008278114178

    Article  Google Scholar 

  3. M.J. Best B. Ding (2000) ArticleTitleA decomposition method for global and local quadratic minimization Journal of Global Optimization 16 133–151 Occurrence Handle10.1023/A:1008328726430

    Article  Google Scholar 

  4. I.M. Bomze G. Danninger (1994) ArticleTitleA finite algorithm for solving general quadratic problems Journal of Global Optimization 4 1–16 Occurrence Handle10.1007/BF01096531

    Article  Google Scholar 

  5. R. Cambini C. Sodini (2002) ArticleTitleA finite algorithm for particular d.c. quadratic programming problem Annals of Operations Research 117 33–49 Occurrence Handle10.1023/A:1021509220392

    Article  Google Scholar 

  6. L. Churilov M. Sniedovich (1999) A concave composite programming perspective on D.C. programming A. Eberhard R. Hill D. Ralph B.M. Glover (Eds) Progress in Optimization Applied Optimization NumberInSeriesVol. 30 Kluwer Academic Publishers Dordrecht

    Google Scholar 

  7. P.L. Angelis ParticleDe P.M. Pardalos G. Toraldo (1997) Quadratic Programming with Box Constraints I.M. Bomze (Eds) et al. Developments in Global Optimization Kluwer Academic Publishers Dordrecht 73–93

    Google Scholar 

  8. A. Ellero (1996) ArticleTitleThe optimal level solutions method Journal of Information and Optimization Sciences 17 IssueID2 355–372

    Google Scholar 

  9. J.E. Falk R.M. Soland (1969) ArticleTitleAn algorithm for separable nonconvex programming problems Management Science 15 IssueID9 550–569 Occurrence Handle10.1287/mnsc.15.9.550

    Article  Google Scholar 

  10. F.R. Gantmacher (1960) The Theory of Matrices NumberInSeriesVol. 1 Chelsea Publishing Company New York

    Google Scholar 

  11. Hiriart-Urruty, J.B. (1985), Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and Duality in Optimization, Lecture Notes in Economics and Mathematical Systems, Vol. 256, Springer-Verlag.

  12. R. Horst (1976) ArticleTitleAn algorithm for nonconvex programming problems Mathematical Programming 10 IssueID3 312–321

    Google Scholar 

  13. R. Horst (1980) ArticleTitleA note on the convergence of an algorithm for nonconvex programming problems Mathematical Programming 19 IssueID2 237–238

    Google Scholar 

  14. R. Horst H. Tuy (1990) Global Optimization Springer-Verlag Berlin

    Google Scholar 

  15. R. Horst P.M. Pardalos N.V. Thoai (1995) Introduction to global optimization. Nonconvex Optimization and Its Applications NumberInSeriesVol. 3 Kluwer Academic Publishers Dordrecht

    Google Scholar 

  16. R. Horst Nguyen Thoai ParticleVan (1996) ArticleTitleA new algorithm for solving the general quadratic programming problem Computational Optimization and Applications 5 IssueID1 39–48 Occurrence Handle10.1007/BF00429750

    Article  Google Scholar 

  17. H. Konno (1976) ArticleTitleMaximization of a convex quadratic function under linear constraints Mathematical Programming 11 IssueID2 117–127

    Google Scholar 

  18. An Thi Hoai ParticleLe Tao Pham Dinh (1997) ArticleTitleSolving a class of linearly constrained indefinite quadratic problems by D.C. algorithms Journal of Global Optimization 11 253–285

    Google Scholar 

  19. L.D. Muu W. Oettli (1991) ArticleTitleAn algorithm for indefinite quadratic programming with convex constraints Operations Research Letters 10 IssueID6 323–327 Occurrence Handle10.1016/0167-6377(91)90004-9

    Article  Google Scholar 

  20. P.M. Pardalos J.H. Glick J.B. Rosen (1987) ArticleTitleGlobal minimization of indefinite quadratic problems Computing 39 IssueID4 281–291

    Google Scholar 

  21. P.M. Pardalos (1991) ArticleTitleGlobal optimization algorithms for linearly constrained indefinite quadratic problems Computers and Mathematics with Applications 21 87–97

    Google Scholar 

  22. Phong Quynh Thai Hoai AnLe Thi Dinh Tao Pham (1995) ArticleTitleDecomposition branch and bound method for globally solving linearly constrained indefinite quadratic minimization problems Operations Research Letters 17 IssueID5 215–220

    Google Scholar 

  23. J.B. Rosen P.M. Pardalos (1986) ArticleTitleGlobal minimization of large scale constrained concave quadratic problems by separable programming Mathematical Programming 34 163–174 Occurrence Handle10.1007/BF01580581

    Article  Google Scholar 

  24. H. Tuy (1995) D.C. optimization: theory, methods and algorithms R. Horst P.M. Pardalos (Eds) Handbook of Global Optimization, Nonconvex Optimization and Its Applications NumberInSeriesVol. 2 Kluwer Academic Publishers Dordrecht 149–216

    Google Scholar 

  25. H. Tuy (1998) Convex analysis and global optimization, Nonconvex Optimization and Its Applications Kluwer Academic Publishers Dordrecht

    Google Scholar 

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Authors and Affiliations

  1. Department of Statistics and Applied Mathematics, University of Pisa, Via Cosimo Ridolfi 10, 56124, Pisa, Italy

    Riccardo Cambini & Claudio Sodini

Authors
  1. Riccardo Cambini
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  2. Claudio Sodini
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Correspondence to Claudio Sodini.

Additional information

*This paper has been partially supported by M.I.U.R. and C.N.R.

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Cambini, R., Sodini, C. Decomposition Methods for Solving Nonconvex Quadratic Programs via Branch and Bound*. J Glob Optim 33, 313–336 (2005). https://doi.org/10.1007/s10898-004-6095-8

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  • DOI: https://doi.org/10.1007/s10898-004-6095-8

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