A new use of Douglas–Rachford splitting for identifying infeasible, unbounded, and pathological conic programs

Abstract

In this paper, we present a method for identifying infeasible, unbounded, and pathological conic programs based on Douglas–Rachford splitting. When an optimization program is infeasible, unbounded, or pathological, the iterates of Douglas–Rachford splitting diverge. Somewhat surprisingly, such divergent iterates still provide useful information, which our method uses for identification. In addition, for strongly infeasible problems the method produces a separating hyperplane and informs the user on how to minimally modify the given problem to achieve strong feasibility. As a first-order method, the proposed algorithm relies on simple subroutines, and therefore is simple to implement and has low per-iteration cost.

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Acknowledgements

W. Yin would like to thank Professor Yinyu Ye for his question regarding ADMM applied to infeasible linear programs during the 2014 Workshop on Optimization for Modern Computation held at Peking University.

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Correspondence to Wotao Yin.

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This work is supported in part by NSF Grant DMS-1720237 and ONR Grant N000141712162.

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Liu, Y., Ryu, E.K. & Yin, W. A new use of Douglas–Rachford splitting for identifying infeasible, unbounded, and pathological conic programs. Math. Program. 177, 225–253 (2019). https://doi.org/10.1007/s10107-018-1265-5

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Keywords

  • Douglas–Rachford splitting
  • Infeasible
  • Unbounded
  • Pathological
  • Conic programs

Mathematics Subject Classification

  • 47H05
  • 65K05
  • 65K15
  • 90C25