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On global optimization with indefinite quadratics

Author

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  • Marcia Fampa

    (Universidade Federal do Rio de Janeiro)

  • Jon Lee

    (University of Michigan)

  • Wendel Melo

    (Universidade Federal do Rio de Janeiro)

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.

Suggested Citation

  • Marcia Fampa & Jon Lee & Wendel Melo, 2017. "On global optimization with indefinite quadratics," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(3), pages 309-337, September.
    • Handle: RePEc:spr:eurjco:v:5:y:2017:i:3:d:10.1007_s13675-016-0079-6
    DOI: 10.1007/s13675-016-0079-6
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    References listed on IDEAS

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    1. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    2. R. Horst & N. V. Thoai, 1999. "DC Programming: Overview," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 1-43, October.
    3. Le Thi Hoai An & Pham Dinh Tao & Le Dung Muu, 1998. "A Combined D.C. Optimization—Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems," Journal of Combinatorial Optimization, Springer, vol. 2(1), pages 9-28, March.
    4. X. Zheng & X. Sun & D. Li, 2011. "Nonconvex quadratically constrained quadratic programming: best D.C. decompositions and their SDP representations," Journal of Global Optimization, Springer, vol. 50(4), pages 695-712, August.
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    Cited by:

    1. R. Cambini & R. Riccardi & D. Scopelliti, 2023. "Solving linear multiplicative programs via branch-and-bound: a computational experience," Computational Management Science, Springer, vol. 20(1), pages 1-32, December.
    2. Marcia Fampa & Jon Lee, 2021. "Convexification of bilinear forms through non-symmetric lifting," Journal of Global Optimization, Springer, vol. 80(2), pages 287-305, June.
    3. Marcia Fampa & Francisco Pinillos Nieto, 2018. "Extensions on ellipsoid bounds for quadratic integer programming," Journal of Global Optimization, Springer, vol. 71(3), pages 457-482, July.
    4. Tiago Andrade & Fabricio Oliveira & Silvio Hamacher & Andrew Eberhard, 2019. "Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming," Journal of Global Optimization, Springer, vol. 73(4), pages 701-722, April.

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