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Journal of Scientific Computing

, Volume 78, Issue 3, pp 1790–1810 | Cite as

An Adaptive Infeasible-Interior-Point Method with the One-Norm Wide Neighborhood for Semi-definite Programming

  • Ximei YangEmail author
  • Yanqin Bai
Article
  • 64 Downloads

Abstract

In this paper, we present a Mehrotra-type predictor–corrector infeasible-interior-point method, based on the one-norm wide neighborhood, for semi-definite programming. The proposed algorithm uses Mehrotra’s adaptive updating scheme for the centering parameter, which incorporates a safeguard strategy that keeps the iterates in a prescribed neighborhood and allows to get a reasonably large step size. Moreover, by using an important inequality that is the relationship between the one-norm and the Frobenius-norm, the convergence is shown for a commutative class of search directions. In particular, the complexity bound is \(\mathcal {O}(n\log \varepsilon ^{-1})\) for Nesterov–Todd direction, and \(\mathcal {O}(n^{3/2}\log \varepsilon ^{-1})\) for Helmberg–Kojima–Monteiro directions, where \(\varepsilon \) is the required precision. The derived complexity bounds coincide with the currently best known theoretical complexity bounds obtained so far for the infeasible semi-definite programming. Some preliminary numerical results are provided as well.

Keywords

Semi-definite programming Wide neighborhood Adaptive updating scheme Infeasible-interior-point method Complexity bound 

Mathematics Subject Classification

65K05 90C22 90C51 

Notes

Acknowledgements

We would like to thank the support of National Natural Science Foundation of China (NNSFC) under Grant Nos. 11501180, 11601134 and 11671122, Chinese Postdoctoral Science Foundation No. 2016M590346, Henan Normal University Doctoral Startup Issues No. qd14150 and Young Scientists Foundation No. 2014QK03, and Innovative Research Team (in Science and Technology) in University of Henan Province No. 14IRTSTHN023.

References

  1. 1.
    Alizadeh, F.: Combinatorial optimization with interior-point methods and semi-definite matrices Ph.D. thesis, Computer Sience Department, University of Minnesota, Minneapolis, USA (1991)Google Scholar
  2. 2.
    Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM J. Optim. 5, 13–51 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Borchers, B.: Sdplib 1.2, a library of semidefinite programming test problems. Optim. Methods Softw. 11, 683–690 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boyd, S., Ghaoui, L.E., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  5. 5.
    De Klerk, E.: Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications. Kluwer Academic Publishers, Dordrecht (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6, 342–361 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991)CrossRefzbMATHGoogle Scholar
  8. 8.
    Ji, J., Potra, F.A., Sheng, R.: On the local convergence of a predictor–corrector method for semi-definite programming. SIAM J. Optim. 10, 195–210 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices. SIAM J. Optim. 7, 86–125 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kojima, M., Shida, M.A., Shindoh, S.: Local convergence of predictor–corrector infeasible-interior-point algorithm for SDPs and SDLCPs. Math. Program. 80, 129–160 (1998)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Koulaei, M.H., Terlaky, T.: On the complexity analysis of a Mehrotra-type primal--dual feasible algorithm for semidefinite optimization. Optim. Methods Softw. 25, 467–485 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, Y., Terlaky, T.: A new class of large neighborhood path-following interior point algorithms for semidefinite optimization with \(O(\sqrt{n}\log {(\text{ Tr }(X^0S^0)/{\varepsilon })})\) iteration complexity. SIAM J. Optim. 20, 2853–2875 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu, H., Liu, C., Yang, X.: New complexity analysis of a Mehrotra-type predictor-corrector algorithm for semidefinite programming. Optim. Methods Softw. 28, 1179–1194 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Luo, Z.Q., Sturm, J.F., Zhang, S.: Conic convex programming and self-dual embedding. Optim. Methods Softw. 14, 169–218 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mehrotra, S.: On the implementation of a primal–dual interior point method. SIAM J. Optim. 2, 575–601 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Monteiro, R.D.C.: Primal–dual path-following algorithms for semidefenite programming. SIAM J. Optim. 7, 663–678 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Monteiro, R.D.C.: Polynomial convergence of primal--dual algorithms for semidefinite programming based on Monteiro and Zhang family of directions. SIAM J. Optim. 8, 797–812 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Monteiro, R.D.C., Zhang, Y.: A unified analysis for a class of long-step primal--dual path-following interior-point algorithms for semidefinite programming. Math. Program. 81, 281–299 (1998)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Nesterov, Y.E., Nemirovsky, A.S.: Interior-Point Methods in Convex Programming: Theory and Applications. SIAM, Philsdephia (1994)Google Scholar
  20. 20.
    Nesterov, Y., Todd, M.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22, 1–42 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nesterov, Y., Todd, M.: Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim. 8, 324–364 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Peng, J., Roos, C., Terlaky, T.: Self-Regularity: An New Paradigm for Primal–Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)zbMATHGoogle Scholar
  23. 23.
    Peng, J., Roos, C., Terlaky, T.: Self-regular functions and new search directions for linear and semidefinite optimization. Math. Program. Ser. A 93, 129–171 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Potra, F.A., Sheng, R.: A superlinearly convergent primal–dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8, 1007–1028 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Potra, F.A., Sheng, R.: On homogeneous interior-point algorithms for semi-definite programming. Optim. Methods Softw. 9, 161–184 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Potra, F.A., Sheng, R.: Superlinear convergence of interior-point algorithms for semidefinite programming. J. Optim. Theory Appl. 99, 103–119 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rangarajan, B.K.: Polynomial convergence of infeasible-interior-point methods over symmetric cones. SIAM J. Optim. 16, 1211–1229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Salahi, M., Mahdavi-Amiri, N.: Polynomial time second order Mehrotra-type predictor–corrector algorithms. Appl. Math. Comput. 183, 646–658 (2006)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Salahi, M., Peng, J., Terlaky, T.: On Mehrotra-type predictor–corrector algorithms. SIAM J. Optim. 18, 1377–1397 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Method Soft. 11, 625–653 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Todd, M.J., Toh, K.C., Tütüncü, R.H.: On the Nesterov–Todd direction in semidefinite programming. SIAM J. Optim. 8, 769–796 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Yang, X., Liu, H., Zhang, Y.: A second-order Mehrotra-type predictor–corrector algorithm with a new wide neighbourhood for semi-definite programming. Inter. J. Comput. Math. 91, 1082–1096 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ye, Y.: A class of projective transformations for linear programming. SIAM J. Comput. 19, 457–466 (1990)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, Y.: On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem. SIAM J. Optim. 4, 208–227 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Zhang, Y.: On extending some primal–dual interior-point algorithms from linear programming to semidefinite programming. SIAM J. Optim. 8, 365–386 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Zhang, Y., Zhang, D.: On polynomial of the Mehrotra-type predictor–corrector interior-point algorithms. Math. Program. 68, 303–318 (1995)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceHenan Normal UniversityXinxiangPeople’s Republic of China
  2. 2.Department of MathematicsShanghai UniversityShanghaiPeople’s Republic of China

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