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EURO Journal on Computational Optimization

, Volume 5, Issue 3, pp 309–337 | Cite as

On global optimization with indefinite quadratics

  • Marcia Fampa
  • Jon Lee
  • Wendel Melo
Original Paper

Abstract

We present an algorithmic framework for global optimization problems in which the non-convexity is manifested as an indefinite-quadratic as part of the objective function. Our solution approach consists of applying a spatial branch-and-bound algorithm, exploiting convexity as much as possible, not only convexity in the constraints, but also extracted from the indefinite-quadratic. A preprocessing stage is proposed to split the indefinite-quadratic into a difference of convex quadratic functions, leading to a more efficient spatial branch-and-bound focused on the isolated non-convexity. We investigate several natural possibilities for splitting an indefinite quadratic at the preprocessing stage, and prove the equivalence of some of them. Through computational experiments with our new solver iquad, we analyze how the splitting strategies affect the performance of our algorithm, and find guidelines for choosing amongst them.

Keywords

Global optimization Indefinite quadratic Difference of convex functions Eigendecomposition Semidefinite programming Mixed-integer non-linear programming 

Mathematics Subject Classification

90-XX 90Cxx 90C26 90C11 90C20 90C22 

Notes

Acknowledgements

The authors gratefully acknowledge the Newton Institute for partial support. J. Lee was partially supported by NSF Grant CMMI-1160915 and ONR Grant N00014-14-1-0315. M. Fampa was partially supported by a Research Grant from CNPq-Brazil. W. Melo contributed much of his work while visiting the University of Michigan, supported by a Research Fellowship from CNPq-Brazil. The authors are also grateful to Sam Burer for providing us the BoxQP instances.

References

  1. Adjiman CS, Dallwig S, Floudas CA, Neumaier A (1998a) A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs—I. Theoretical advances. Comput Chem Eng 22(9):1137–1158CrossRefGoogle Scholar
  2. Adjiman CS, Dallwig S, Floudas CA, Neumaier A (1998b) A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs—II. Implementation and computational results. Comput Chem Eng 22(9):1159–1179CrossRefGoogle Scholar
  3. Alizadeh F (1993) Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J Optim 5:13–51CrossRefGoogle Scholar
  4. An LTH, Tao PD, Muu LD (1998) A combined d.c. optimization—ellipsoidal branch-and-bound algorithm for solving nonconvex quadratic programming problems. Semidefinite programming and interior-point approaches for combinatorial optimization problems (Toronto, ON, 1996). J Comb Optim 2(1):9–28CrossRefGoogle Scholar
  5. An LTH, Tao PD (2005) The DC (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann Oper Res 133:23–46CrossRefGoogle Scholar
  6. Anstreicher KM, Burer S (2005) D.C. versus copositive bounds for standard QP. J Glob Optim 33:299–312CrossRefGoogle Scholar
  7. Belotti P, Lee J, Liberti L, Margot F, Wächter A (2009) Branching and bounds tightening techniques for non-convex MINLP. Optim Methods Softw 24:597–634CrossRefGoogle Scholar
  8. Billionnet A, Elloumi S, Lambert A (2014) A branch and bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation. J Comb Optim 2(28):376–399CrossRefGoogle Scholar
  9. Bomze IM, Dür MD, De Klerk E, Roos C, Quist AJ, Terlaky T (2000) On copositive programming and standard quadratic optimization problems. J Glob Optim 18:301–320CrossRefGoogle Scholar
  10. Bomze IM, Locatelli M (2004) Undominated d.c. decompositions of quadratic functions and applications to branch-and-bound approaches. Comput Optim Appl 28(2):227–245CrossRefGoogle Scholar
  11. Bradley SP, Hax AC, Magnanti TL (1977) Applied mathematical programming. Addison-Wesley, BostonGoogle Scholar
  12. Burer S, Chen J (2012) Globally solving nonconvex quadratic programming problems via completely positive programming. Math Program Comput 4:33–52CrossRefGoogle Scholar
  13. Burer S, Saxena A (2012) The MILP road to MIQCP. In: Lee J, Leyffer S (eds) Mixed-integer nonlinear programming. The IMA volumes in mathematics and its applications, vol 154. Springer, New York, pp 373–405Google Scholar
  14. Burer S, Vandenbussche D (2008) A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math Program Ser A 113:259–282CrossRefGoogle Scholar
  15. Burkard RE, Çela E, Pardalos PM, Pitsoulis LS (1998) The quadratic assignment problem. Handbook of combinatorial optimization, vol 3. Kluwer Acad. Publ., Boston, pp 241–237Google Scholar
  16. Calamai PH, Vicente LN, Júdice JJ (1993) A new technique for generating quadratic programming test problems. Math Program 61:215–231CrossRefGoogle Scholar
  17. D’Ambrosio C, Lee J, Wächter A (2012) An algorithmic framework for MINLP with separable non-convexity. In: Lee J, Leyffer S (eds) Mixed-integer nonlinear programming. The IMA volumes in mathematics and its applications, vol 154, pp 315–347Google Scholar
  18. Hemmecke R, Köppe M, Lee J, Weismantel R (2010) Nonlinear integer programming. In: Jünger M, Liebling T, Naddef D, Nemhauser G, Pulleyblank W, Reinelt G, Rinaldi G, Wolsey L (eds) 50 Years of integer programming 1958–2008: the early years and state-of-the-art surveys. Springer, Berlin, pp 561–618Google Scholar
  19. Horst R, Thoai NV (1999) DC programming: overview. J Optim Theory Appl 103:1–43CrossRefGoogle Scholar
  20. Lee J (2007) In situ column generation for a cutting-stock problem. Comput Oper Res 34(8):2345–2358CrossRefGoogle Scholar
  21. Lee J, Leyffer S (eds) (2012) Mixed-integer nonlinear programming. The IMA volumes in mathematics and its applications, vol 154. Springer, New YorkGoogle Scholar
  22. Rendl F, Rinaldi G, Wiegele A (2010) Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math Program Ser A 121(2):307–335CrossRefGoogle Scholar
  23. Saxena A, Bonami P, Lee J (2010) Convex relaxations of non-convex mixed-integer quadratically-constrained programs: extended formulations. Math Program Ser B 124(1–2):383–411CrossRefGoogle Scholar
  24. Saxena A, Bonami P, Lee J (2011) Convex relaxations of non-convex mixed-integer quadratically-constrained programs: projected formulations. Math Program Ser A 130(2):359–413CrossRefGoogle Scholar
  25. Tao PD, An LTH (1996) d.c. (difference of convex functions) optimization algorithms (DCA) for globally minimizing nonconvex quadratic forms on Euclidean balls and spheres. Oper Res Lett 19:207–216CrossRefGoogle Scholar
  26. Tawarmalani M, Sahinidis NV (2005) A polyhedral branch-and-cut approach to global optimization. Math Program 103(2):225–249CrossRefGoogle Scholar
  27. Zheng XJ, Sun XL, Li D (2011) Nonconvex quadratically constrained quadratic programming: best d.c. decompositions and their SDP representations. J Glob Optim 50:695–712CrossRefGoogle Scholar

Copyright information

© EURO - The Association of European Operational Research Societies 2016

Authors and Affiliations

  1. 1.Universidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.University of MichiganAnn ArborUSA

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