On global optimization with indefinite quadratics
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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.
KeywordsGlobal optimization Indefinite quadratic Difference of convex functions Eigendecomposition Semidefinite programming Mixed-integer non-linear programming
Mathematics Subject Classification90-XX 90Cxx 90C26 90C11 90C20 90C22
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.
- 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
- Bradley SP, Hax AC, Magnanti TL (1977) Applied mathematical programming. Addison-Wesley, BostonGoogle Scholar
- 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
- 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
- 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
- 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
- Lee J, Leyffer S (eds) (2012) Mixed-integer nonlinear programming. The IMA volumes in mathematics and its applications, vol 154. Springer, New YorkGoogle Scholar