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A simplified treatment of Ramana’s exact dual for semidefinite programming

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In semidefinite programming the dual may fail to attain its optimal value and there could be a duality gap, i.e., the primal and dual optimal values may differ. In a striking paper, Ramana (Math. Program. Ser. B 77, 129–162, 1997) proposed a polynomial size extended dual that does not have these deficiencies and yields a number of fundamental results in complexity theory. In this work we walk the reader through a concise and self-contained derivation of Ramana’s dual, relying mostly on elementary linear algebra.

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Fig. 1

Notes

  1. Proposition 3 is usually stated in the space \({\mathbb {R}}^{n}.\)

  2. That is, K is closed, convex, and \(\lambda x \in K\) for all \(x \in K\) and \(\lambda \ge 0.\)

  3. This means two things: (i) it is a convex subset of \({{\mathcal {S}}}_+^{n}\) and (ii) if X, Y are in \({{\mathcal {S}}}_+^{n},\) and the open line segment \(\{ \lambda X + (1-\lambda )Y : \lambda \in (0,1) \}\) intersects \(F, \, \) then both X and Y must be in F.

  4. Afterwards \(U_1\) may not look like in Eq. (3.39) anymore, but for our purposes this does not matter.

  5. The following argument may better explain the role of the \(Y_i\) matrices. If S is any slack, then \(\langle S, Y_1 \rangle = \langle S, Y_2 \rangle = 0. \, \) We invite the reader to check that these equations lead to the same argument that we gave in paragraph (1) that show the last two rows and columns of S are zero.

  6. We can use any matrix norm, for example the spectral norm or the Frobenius norm.

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Acknowledgements

Part of this paper was written during a visit to Chapel Hill by the first author and he would like to express his gratitude to Prof. Takashi Tsuchiya, for creating conditions that made this visit possible. The work of the first author was partially supported by the JSPS KAKENHI Grant Numbers JP15H02968 and JP19K20217. The second author thanks Pravesh Kothari and Ryan O’ Donnell for helpful discussions on SDP. The work of the second author was supported by the National Science Foundation, award DMS-1817272. Both authors are grateful to Siyuan Chen, Alex Touzov, and Yuzixuan Zhu for their careful reading of the manuscript and their helpful comments. Most importantly, we are very grateful to the anonymous referees whose comments and suggestions greatly improved the manuscript.

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  1. Department of Statistical Inference and Mathematics, Institute of Statistical Mathematics, Tachikawa, Tokyo, Japan

    Bruno F. Lourenço

  2. Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, USA

    Gábor Pataki

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  1. Bruno F. Lourenço
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  2. Gábor Pataki
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Correspondence to Gábor Pataki.

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Lourenço, B.F., Pataki, G. A simplified treatment of Ramana’s exact dual for semidefinite programming. Optim Lett 17, 219–243 (2023). https://doi.org/10.1007/s11590-022-01898-2

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