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Convex Relaxations and Integrality Gaps

In: Handbook on Semidefinite, Conic and Polynomial Optimization

Author

Listed:
  • Eden Chlamtac

    (Tel Aviv University)

  • Madhur Tulsiani

    (Toyota Technological Institute at Chicago)

Abstract

We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovasz and Schrijver, Sherali and Adams, and Lasserre generate increasingly strong linear and semidefinite programming relaxations starting from a basic one. We survey some positive applications of these hierarchies, where their use yields improved approximation algorithms. We also discuss known lower bounds on the integrality gaps of relaxations arising from these hierarchies, demonstrating limits on the applicability of such hierarchies for certain optimization problems.

Suggested Citation

  • Eden Chlamtac & Madhur Tulsiani, 2012. "Convex Relaxations and Integrality Gaps," International Series in Operations Research & Management Science, in: Miguel F. Anjos & Jean B. Lasserre (ed.), Handbook on Semidefinite, Conic and Polynomial Optimization, chapter 0, pages 139-169, Springer.
    • Handle: RePEc:spr:isochp:978-1-4614-0769-0_6
    DOI: 10.1007/978-1-4614-0769-0_6
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    Cited by:

    1. Adam Kurpisz & Samuli Leppänen & Monaldo Mastrolilli, 2017. "On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 135-143, January.
    2. Vivek Bagaria & Jian Ding & David Tse & Yihong Wu & Jiaming Xu, 2020. "Hidden Hamiltonian Cycle Recovery via Linear Programming," Operations Research, INFORMS, vol. 68(1), pages 53-70, January.
    3. Jean B. Lasserre & Kim-Chuan Toh & Shouguang Yang, 2017. "A bounded degree SOS hierarchy for polynomial optimization," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(1), pages 87-117, March.
    4. Kyeong Soo Kim & Sanghyuk Lee & Tiew On Ting & Xin-She Yang, 2019. "Atomic Scheduling of Appliance Energy Consumption in Residential Smart Grids," Energies, MDPI, vol. 12(19), pages 1-18, September.

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