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Strict Feasibility of Conic Optimization Problems

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A Mathematical Approach to Research Problems of Science and Technology

Part of the book series: Mathematics for Industry ((MFI,volume 5))

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Abstract

A conic optimization problem (COP) is the problem of minimizing a given linear objective function over the intersection of an affine space and a closed convex cone. Conic optimization problem is often used for solving nonconvex optimization problems. The strict feasibility of COP is important from the viewpoint of computation. The lack of the strict feasibility may cause the instability of computation. This article provides a brief introduction of COP and a characterization of the strict feasibility of COP. We also explain a facial reduction algorithm (FRA), which is based on the characterization. This algorithm can generate a strictly feasible COP which is equivalent to the original COP, or detect the infeasibility of COP.

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Notes

  1. 1.

    We remark that the formulation (3) of SDP in this article is different from one in the article of Dr. Fujisawa. However, one can obtain the form in the article of Dr. Fujisawa by applying the following replacement:

    $$ \begin{array}{cc} b \rightarrow -c,&L_i \rightarrow -F_i. \end{array} $$

    Then, (3) can be reformulated as a minimization problem on \(y\).

  2. 2.

    For a given convex set \(C\subset \mathbb {R}^n\), a face of \(C\) is a convex subset \(D\) of \(C\) such that every \(x, y\in C\), \(x+y\in D\) implies that \(x, y\in D\). If \(C\) is polyhedral, then the definition is simpler. In fact, a face of polyhedral set \(C\) is the intersection with a hyperplane and \(C\). Such a face is called exposed face. See [10, 14] for more details.

References

  1. A. Ben-Tal, A. Nemirovski, Lectures on Modern Convex Optimization (Society for Industrial and Applied Mathematics, Philadelphia, 2001)

    Google Scholar 

  2. M.J. Borwein, H. Wolkowicz, Facial reduction for a cone-convex programming problem. J. Aust. Math. Soc. 30, 369–380 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Burer, On the copositive representation of binary and continuous non convex quadratic programs. Math. Programm. 120, 479–495 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Burer, in Copositive Programming, ed. by M.F. Anjos, J.B. Lasserre. Handbook on Semidefinite, Conic and Polynomial Optimization (Springer, New York, 2012), pp 201–218

    Google Scholar 

  5. F. Burkowski, Y. Cheung, H. Wolkowicz, Efficient use of semidefinite programming for selection of rotamers in protein conformations, preprint (2013)

    Google Scholar 

  6. N. Krislock, H. Wolkowicz, Explicit sensor network localization using semidefinite representations and facial reductions. SIAM J. Optim. 20, 2679–2708 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Kojima, S. Kim, H. Waki, Sparsity in sums of squares of polynomials. Math. Program. 103, 45–62 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Lobo, L. Vandenberghe, S. Boyd, H. Lebret, Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Z.-Q. Luo, F.J. Sturm, S. Zhang, Duality results for conic convex programming, Econometric institute report no. 9719/A. Econometric Institute, Erasmus University Rotterdam (1997)

    Google Scholar 

  10. G. Pataki, in The Geometry of Cone-LP’s, ed. by H. Wolkowicz, R. Saigal, L. Vandenberghe. The Handbook of Semidefinite Programming (Springer, Berlin, 2000), pp. 29–65

    Google Scholar 

  11. G. Pataki, Strong duality in conic linear programming: facial reduction and extended dual, arXiv:1301.7717 (2013)

  12. V.M. Ramana, An exact duality theory for semidefinite programming and its complexity implications. Math. Program. 77, 129–162 (1997)

    MathSciNet  MATH  Google Scholar 

  13. V.M. Ramana, L. Tunçel, H. Wolkowicz, Strong duality for semidefinite programming. SIAM J. Optim. 7, 641–662 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. R.T. Rockafellar, in Convex Analysis. Princeton Landmarks in Mathematics and Physics (Princeton University Press, Princeton, 1970)

    Google Scholar 

  15. A. Schrijver, Theory of Linear and Integer Programming (Wiley, New York, 1979)

    Google Scholar 

  16. F.J. Sturm, Primal-Dual Interior Point Approach to Semidefinite Programming, Ph.D. Thesis, Erasmus University Rotterdam (1997)

    Google Scholar 

  17. F.J. Sturm, in Theory and Algorithms of Semidefinite Programming, ed. by H. Frenk, K. Roos, T. Terlaky. High performance optimization, pp. 1–194 (Kluwer Academic Publishers, Dordrecht, 2000)

    Google Scholar 

  18. M.J. Todd, Semidefinite optimization. Acta Numerica 10, 515–560 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. L. Tunçel, On the Slater condition for the SDP relaxations of nonconvex sets. Oper. Res. Lett. 29, 181–186 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Tunçel, in Polyhedral and Semidefinite Programming Methods in Combinatorial Optimization. Fields Institute Monographs (2010)

    Google Scholar 

  21. L. Tunçel, H. Wolkowicz, Strong duality and minimal representations for cone optimization. Comput. Optim. Appl. 53, 619–648 (2013)

    Google Scholar 

  22. H. Waki, M. Muramatsu, Facial reduction algorithms for finding sparse SOS representations. Oper. Res. Lett. 38, 361–365 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. H. Waki, M. Muramatsu, An extension of the elimination method for a sparse SOS polynomial. J. Oper. Res. Soc. Jpn 54, 161–190 (2011)

    MathSciNet  MATH  Google Scholar 

  24. H. Waki, M. Muramatsu, Facial reduction algorithms for conic optimization problems. J. Optim. Theor. Appl. 158, 188–215 (2013)

    Google Scholar 

  25. H. Waki, M. Nakata, M. Muramatsu, Strange behaviors of interior-point methods for solving semidefinite programming problems in polynomial optimization. Comput. Optim. Appl. 53, 823–844 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. H. Wolkowicz, Q. Zhao, Semidefinite programming relaxations for the graph partitioning problem. Discrete Appl. Math. 96–97, 461–479 (1999)

    Article  MathSciNet  Google Scholar 

  27. Q. Zhao, S.E. Karisch, F. Rendl, H. Wolkowicz, Semidefinite programming relaxations for the quadratic assignment problem. J. Comb. Optim. 2, 71–109 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author was supported by a Grant-in-Aid for JSPS Fellow 20003236 and a Grant-in-Aid for Young Scientists (B) 22740056.

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Authors and Affiliations

  1. Institute of Mathematics for Industry, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Hayato Waki

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  1. Hayato Waki
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Correspondence to Hayato Waki .

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Editors and Affiliations

  1. Kyushu University, Fukuoka, Japan

    Ryuei Nishii

  2. Department of Mathematics, Hokkaido University, Sapporo, Japan

    Shin-ichiro Ei

  3. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Miyuki Koiso

  4. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Hiroyuki Ochiai

  5. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Kanzo Okada

  6. Faculty of Arts and Science, Kyushu University, Fukuoka, Japan

    Shingo Saito

  7. Kyushu University Institute of Mathematics for Industry, Fukuoka, Japan

    Tomoyuki Shirai

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Waki, H. (2014). Strict Feasibility of Conic Optimization Problems. In: Nishii, R., et al. A Mathematical Approach to Research Problems of Science and Technology. Mathematics for Industry, vol 5. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55060-0_24

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  • DOI: https://doi.org/10.1007/978-4-431-55060-0_24

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