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Strong duality and minimal representations for cone optimization

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Abstract

The elegant theoretical results for strong duality and strict complementarity for linear programming, LP, lie behind the success of current algorithms. In addition, preprocessing is an essential step for efficiency in both simplex type and interior-point methods. However, the theory and preprocessing techniques can fail for cone programming over nonpolyhedral cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, CQ, and strict complementarity, for linear cone optimization problems in finite dimensions. One theme is the notion of minimal representation of the cone and the constraints. This provides a framework for preprocessing cone optimization problems in order to avoid both the theoretical and numerical difficulties that arise due to the (near) loss of the strong CQ, strict feasibility. We include results and examples on the surprising theoretical connection between duality gaps in the original primal-dual pair and lack of strict complementarity in their homogeneous counterpart. Our emphasis is on results that deal with Semidefinite Programming, SDP.

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Notes

  1. Similarly, we can use the (2,2) position rather than the (3,3) position.

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Acknowledgements

Research supported by The Natural Sciences and Engineering Research Council of Canada. Part of this paper is based on work presented at: ICCOPT II & MOPTA-07, McMaster University, Hamilton, 2007; 9th SIAM Conference on Optimization, Boston, May, 2008; and HPOPT 10, Tilburg, 2008. The authors would like to acknowledge Simon Schurr (Univ. of Waterloo) for many helpful conversations, as well as for other contributions to this paper. URL for paper: orion.math.uwaterloo.ca/~hwolkowi/henry/reports/ABSTRACTS.html

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  1. Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Levent Tunçel & Henry Wolkowicz

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Tunçel, L., Wolkowicz, H. Strong duality and minimal representations for cone optimization. Comput Optim Appl 53, 619–648 (2012). https://doi.org/10.1007/s10589-012-9480-0

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