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Partial Lagrangian relaxation for the unbalanced orthogonal Procrustes problem

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  • Yong Xia
  • Ying-Wei Han

Abstract

Based on a novel reformulation of the feasible region, we propose and analyze a partial Lagrangian relaxation approach for the unbalanced orthogonal Procrustes problem (UOP). With a properly selected Lagrangian multiplier, the Lagrangian relaxation (LR) is equivalent to the recent matrix lifting semidefinite programming relaxation (MSDR), which has much more variables and constraints. Numerical results show that (LR) is solved more efficiently than (MSDR). Moreover, based on the special structure of (LR), we successfully employ the well-known Frank–Wolfe algorithm to efficiently solve very large instances of (LR). The rate of the convergence is shown to be independent of the row-dimension of the matrix variable of (UOP). Finally, motivated by (LR), we propose a Lagrangian heuristic for (UOP). Numerical results show that it can efficiently find the global optimal solutions of some randomly generated instances of (UOP). Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Yong Xia & Ying-Wei Han, 2014. "Partial Lagrangian relaxation for the unbalanced orthogonal Procrustes problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 79(2), pages 225-237, April.
    • Handle: RePEc:spr:mathme:v:79:y:2014:i:2:p:225-237
    DOI: 10.1007/s00186-013-0460-7
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    References listed on IDEAS

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    1. Helmberg, C., 2002. "Semidefinite programming," European Journal of Operational Research, Elsevier, vol. 137(3), pages 461-482, March.
    2. Miguel F. Anjos, 2004. "On semidefinite programming relaxations for the satisfiability problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 60(3), pages 349-367, December.
    3. Peter Schönemann, 1966. "A generalized solution of the orthogonal procrustes problem," Psychometrika, Springer;The Psychometric Society, vol. 31(1), pages 1-10, March.
    4. Yichuan Ding & Dongdong Ge & Henry Wolkowicz, 2011. "On Equivalence of Semidefinite Relaxations for Quadratic Matrix Programming," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 88-104, February.
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    Cited by:

    1. Samuel Burer & Kyungchan Park, 2024. "A Strengthened SDP Relaxation for Quadratic Optimization Over the Stiefel Manifold," Journal of Optimization Theory and Applications, Springer, vol. 202(1), pages 320-339, July.

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