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Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization

Author

Listed:
  • Goran Banjac

    (ETH Zurich)

  • Paul Goulart

    (University of Oxford)

  • Bartolomeo Stellato

    (Massachusetts Institute of Technology)

  • Stephen Boyd

    (Stanford University)

Abstract

The alternating direction method of multipliers is a powerful operator splitting technique for solving structured optimization problems. For convex optimization problems, it is well known that the algorithm generates iterates that converge to a solution, provided that it exists. If a solution does not exist, then the iterates diverge. Nevertheless, we show that they yield conclusive information regarding problem infeasibility for optimization problems with linear or quadratic objective functions and conic constraints, which includes quadratic, second-order cone, and semidefinite programs. In particular, we show that in the limit the iterates either satisfy a set of first-order optimality conditions or produce a certificate of either primal or dual infeasibility. Based on these results, we propose termination criteria for detecting primal and dual infeasibility.

Suggested Citation

  • Goran Banjac & Paul Goulart & Bartolomeo Stellato & Stephen Boyd, 2019. "Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 490-519, November.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:2:d:10.1007_s10957-019-01575-y
    DOI: 10.1007/s10957-019-01575-y
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    References listed on IDEAS

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    1. Brendan O’Donoghue & Eric Chu & Neal Parikh & Stephen Boyd, 2016. "Conic Optimization via Operator Splitting and Homogeneous Self-Dual Embedding," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1042-1068, June.
    2. Didier Henrion & Jérôme Malick, 2012. "Projection Methods in Conic Optimization," 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 565-600, Springer.
    3. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
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    Cited by:

    1. Alberto Marchi, 2022. "On a primal-dual Newton proximal method for convex quadratic programs," Computational Optimization and Applications, Springer, vol. 81(2), pages 369-395, March.
    2. Nikitas Rontsis & Paul Goulart & Yuji Nakatsukasa, 2022. "Efficient Semidefinite Programming with Approximate ADMM," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 292-320, January.
    3. Michael Garstka & Mark Cannon & Paul Goulart, 2021. "COSMO: A Conic Operator Splitting Method for Convex Conic Problems," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 779-810, September.
    4. Rui-Jin Zhang & Xin-Wei Liu & Yu-Hong Dai, 2023. "IPRQP: a primal-dual interior-point relaxation algorithm for convex quadratic programming," Journal of Global Optimization, Springer, vol. 87(2), pages 1027-1053, November.

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