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Faster, but weaker, relaxations for quadratically constrained quadratic programs

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Abstract

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.

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Authors and Affiliations

  1. Department of Management Sciences, University of Iowa, Iowa City, IA, 52242-1994, USA

    Samuel Burer

  2. Department of Mathematics, Ewha W. University, 11-1 DaHyun-Dong, SuhDaeMoon-Gu, Seoul, Korea, 120-750

    Sunyoung Kim

  3. Research and Development Initiative & JST CREST, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Masakazu Kojima

Authors
  1. Samuel Burer
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  2. Sunyoung Kim
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  3. Masakazu Kojima
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Corresponding author

Correspondence to Samuel Burer.

Additional information

The research of S. Burer was supported in part by NSF Grant CCF-0545514. The research of S. Kim was supported by NRF 2012-R1A1A2-038982 and NRF 2010-000-8784. The research of M. Kojima was partially supported by the Japan Science and Technology Agency (JST), the Core Research of Evolutionary Science and Technology (CREST) Research Project.

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Burer, S., Kim, S. & Kojima, M. Faster, but weaker, relaxations for quadratically constrained quadratic programs. Comput Optim Appl 59, 27–45 (2014). https://doi.org/10.1007/s10589-013-9618-8

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  • DOI: https://doi.org/10.1007/s10589-013-9618-8

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