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Weak infeasibility in second order cone programming

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Abstract

The objective of this work is to study weak infeasibility in second order cone programming. For this purpose, we consider a sequence of feasibility problems which mostly preserve the feasibility status of the original problem. This is used to show that for a given weakly infeasible problem at most m directions are needed to get arbitrarily close to the cone, where m is the number of Lorentz cones. We also tackle a closely related question and show that given a bounded optimization problem satisfying Slater’s condition, we may transform it into another problem that has the same optimal value but it is ensured to attain it. From solutions to the new problem, we discuss how to obtain solution to the original problem which are arbitrarily close to optimality. Finally, we discuss how to obtain finite certificate of weak infeasibility by combining our own techniques with facial reduction. The analysis is similar in spirit to previous work by the authors on SDPs, but a different approach is required to obtain tighter bounds on the number of directions needed to approach the cone.

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Notes

  1. Recall that if \(x,y \in {\mathcal {K}^{n}}\) satisfy \(x^Ty = 0\), then \(x_0\overline{y} + y_0\overline{x} = 0\).

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Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. M. Muramatsu and T. Tsuchiya are supported in part with Grant-in-Aid for Scientific Research (B)24310112 and (C)2630025. M. Muramatsu is also supported in part with Grant-in-Aid for Scientific Research (B) 26280005.

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Authors and Affiliations

  1. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-41 Ookayama, Meguro-ku, Tokyo, 152-8552, Japan

    Bruno F. Lourenço

  2. Department of Computer Science, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo, 182-8585, Japan

    Masakazu Muramatsu

  3. National Graduate Institute for Policy Studies, 7-22-1 Roppongi, Minato-ku, Tokyo, 106-8677, Japan

    Takashi Tsuchiya

Authors
  1. Bruno F. Lourenço
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  2. Masakazu Muramatsu
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  3. Takashi Tsuchiya
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Correspondence to Bruno F. Lourenço.

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Lourenço, B.F., Muramatsu, M. & Tsuchiya, T. Weak infeasibility in second order cone programming. Optim Lett 10, 1743–1755 (2016). https://doi.org/10.1007/s11590-015-0982-4

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  • DOI: https://doi.org/10.1007/s11590-015-0982-4

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