79
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

A primal–dual predictor–corrector interior-point method for symmetric cone programming with O(√r log ϵ−1) iteration complexity

, &
Pages 1998-2010 | Received 10 Apr 2016, Accepted 17 Nov 2016, Published online: 31 Jan 2017

References

  • W. Ai, Neighborhood-following algorithms for linear programming, Sci. China Ser. A 47 (2004), pp. 812–820.
  • W. Ai and S. Zhang, An O(nL) iteration primal-dual path-following method, based on wide neighborhoods and large updates, for monotone LCP, SIAM J. Optim. 16 (2005), pp. 400–417.
  • Y.Q. Bai, G.Q. Wang, and C. Roos, Primal-dual interior-point algorithms for second-order cone optimization based on kernel functions, Nonlinear Anal. 70 (2009), pp. 3584–3602.
  • J. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, New York, 1994.
  • L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity 1(4) (1997), pp. 331–357.
  • L. Faybusovich, Linear systems in Jordan algebras and primal–dual interior-point algorithms, J. Comput. Appl. Math. 86(1) (1997), pp. 149–175.
  • L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z. 239(1) (2002), pp. 117–129.
  • Z.Z. Feng and L. Fang, A wide neighborhood interior-point method with O(nL) iteration-complexity bound for semidefinite programming, Optimization. 59(8) (2010), pp. 1235–1246.
  • Z.Z. Feng, L. Fang, and G. He, An O(nL) iteration primal-dual path-following method, based on wide neighborhood and large update, for second-order cone programming, Optimization. 63(5) (2014), pp. 679–691.
  • G. Gu, M. Zangiabadi, and C. Roos, Full Nesterov–Todd step interior-point methods for symmetric optimization, Eur. J. Oper. Res. 214(3) (2011), pp. 473–484.
  • O. Guler, Barrier functions in interior-point methods, Math. Oper. Res. 21(4) (1996), pp. 860–885.
  • M. Kojima, S. Mizuno, and A. Yoshise, A Prima–dual interior point algorithm for linear programming, in Progress in Mathematical Programming, N. Megiddo, ed., Springer Verlag, Berlin, 1989.
  • Y. Li and T. Terlaky, A new class of large neighborhood path-following interior point algorithms for semidefinite optimization with O(nlog⁡(Tr(X0S0)/ε)) iteration complexity, SIAM J. Optim. 20 (2010), pp. 2853–2875.
  • C. Liu, Study on complexity of some interior-point algorithms in conic programming, Ph.D. thesis, Xidian University in Chinese, 2012.
  • C. Liu, H. Liu, and X. Liu, Polynomial convergence of second-order Mehrotra-type predictor–corrector algorithms over symmetric cones, J. Optim. Theory Appl. 154(3) (2012), pp. 949–965.
  • H. Liu, X. Yang, and C. Liu, A new wide neighborhood primal-dual infeasible-interior-point method for symmetric cone programming, J. Optim. Theory Appl. 158(3) (2013), pp. 796–815.
  • Z. Luo and N. Xiu, Path-following interior point algorithms for the Cartesian P∗(κ)-LCP over symmetric cones, Sci. China Ser. A 52(8) (2009), pp. 1769–1784.
  • H. Mansouri and C. Roos, Simplifled O(nL) infeasible interior-point algorithm for linear optimization using full-Newton steps, Optim. Methods Softw. 22(3) (2007), pp. 519–530.
  • S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM J. Optim. 2(4) (1992), pp. 575–601.
  • S. Mizuno, M.J. Todd, and Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Math. Oper. Res. 18(4) (1993), pp. 964–981.
  • Y. Nesterov and M. Todd, Self-scaled barriers and interior-point methods for convex programming, Math. Oper. Res. 22(1) (1997), pp. 1–42.
  • Y.E. Nesterov and M.J. Todd, Primal-dual interior-point methods for self-scaled cones, SIAM J. Optim. 8 (1998), pp. 324–364.
  • M. Salahi, J. Peng, and T. Terlaky, On Mehrtora type predictor–corrector algorithms, SIAM J. Optim. 18(4) (2007), pp. 1377–1397.
  • S.H. Schmieta and F. Alizadeh, Extension of primal–dual interior-point algorithms to symmetric cones, Math. Program. 96(3) (2003), pp. 409–438.
  • M.J. Todd, K.C. Toh, and R.H. Tautauncau, On the Nesterov–Todd direction in semidefinite programming, SIAM J. Optim. 8(3) (1998), pp. 769–796.
  • G.Q. Wang and Y.Q. Bai, A new primal–dual path-following interior-point algorithm for semidefinite optimization, J. Math. Anal. Appl. 353 (2009), pp. 339–349.
  • G.Q. Wang and Y.Q. Bai, A new full Nesterov–Todd step primal-dual path-following interior-point algorithm for symmetric optimization, J. Optim. Theory Appl. 154 (2012), pp. 966–985.
  • X. Yang, Y. Zhang, H. Liu, and P. Shen, A new second-order infeasible primal–dual path-following algorithm for symmetric cone optimization, Numer. Funct. Anal. Optim., doi: 10.1080/01630563.2016.1138127.
  • M. Zangiabadi, G. Gu, and C. Roos, A full Nesterov–Todd step infeasible interior-point method for second-order cone optimization, J. Optim. Theory Appl. 158(3) (2013), pp. 816–858.
  • J. Zhang and K. Zhang, Polynomiality complexity of an interior point algorithm with a second order corrector step for symmetric cone programming, Math. Method Oper. Res. 73 (2011), pp. 75–90.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.