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An inexact non-interior continuation method for semidefinite programming: convergence analysis and numerical results

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Abstract

In this paper, an inexact non-interior continuation method is proposed for semidefinite Programs. By a matrix mapping, the primal-dual optimal condition can be inverted into a smoothed nonlinear system of equations. A linear system of equations with residual vector is eventually driven by solving the smoothed nonlinear system of equations and finally solved by the conjugate residual method. The global and locally superlinear convergence are verified. Numerical results and comparisons indicate that the proposed methods are very promising and comparable to several interior-point and other exact non-interior continuation methods.

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References

  1. Alizadeh F.: Interior point methods in semidefinite programming with application to combinatorial optimization. SIAM J. Optim. 5, 13–51 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alizadeh, F., Haeberly, J.-P., Overton M.L.: Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results. SIAM J. Optim. 8, 746–68 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellavia, S., Pieraccini, S.: Convergence Analysis of an inexact infeasible interior point mehtod for semidefinite programming. Comput. Optim. Appl. 29, 289–313 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bartels, R.H., Stewart, G.W.: Solution of the Matrix Equation AX + XB = C. Communications of the ACM 15(9), 820–826 (1972)

    Article  Google Scholar 

  5. Benson, S., Ye, Y., Zhang, X.: Solving large-scale sparse semidefinite programs for combinatorial optimization, vol. 10, pp. 443–461 (2000)

  6. Borchers, B.: SDPLIB 1.2, A Library of Semidefinite Programming Test Problems. Optim. Methods Softw. 11, 597–611 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, X., Qi, H., Tseng, P.: Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems. SIAM J. Optim. 13, 960–985 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, X., Tseng, P.: Non-interior continuation methods for solving semidefinite complementarity problems. Math. Program. 95, 431–474 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Helmberg, C., Rendl, F., Vanderbei, R.J., Wolkowicz, H.: An interior-point method for semidefinite programming. SIAM J. Optim. 6, 342–361 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Golub, G.H., Nash, S., Van Loan, C.F.: A Hessenberg-Schur method for the problem AX + XB = C. IEEE Trans. Auto. Contr. AC-24, 909–913 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kanzow, C., Nagel, C.: Semidefinite programs: new search directions, smoothing-type methods, and numerical results. SIAM J. Optim. 13, 1–23 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kanzow, C., Nagel, C.: Some Practical Aspects of a Newton-Type Method for Semidefinite Programs. Preprint 243. Institute of Applied Mathematics and Statistics, Germany (2001)

    Google Scholar 

  13. Kojima, M., Shida, M., Shindoh, S.: Local convergence of predictor-corrector infeasible interior-point algorithms for SDPs and SDLCPs, vol. 80, pp. 129–160 (1998)

  14. Kojima, M., Shindoh, S., Hara, S.: Interior-point methods for the monotone semidefinite linear complementarity problems. SIAM J. Optim. 7, 86–125 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Monteiro, R.D.C.: Primal-dual path following algorithms for semidefinite programming. SIAM J. Optim. 7, 663–678 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nesterov, Y.E., Nemirovsky, A.: Interior-point polynomial methods in convex programming. Volume 13 of Studies in Applied Mathematics SIAM, Philadelphia, PA (1994)

  17. Nesterov, Y.E., Todd, M.J.: Self-scaled barriers and interior-point methods for convex programming. Math Oper. Res. 22, 1–42 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nesterov, Y.E., Todd, M.J.: Primal-dual interior-point methods for self-scaled cones. SIAM J. Optim. 8, 324–364 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nakata, K., Zhang, S.-L., Kojima, M.: Preconditioned Conjugate Gradient Methods for Large Scale and Dense Linear Systems in Semidefinite Programming, abstract based on talks delivered at INFORMS Meeting. Philadelphia (1999)

  20. Penzl, T.: Numerical solution of generalized Lyapunov equations. Adv. Comput. Math. 8, 33–48 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Qi, L.: Convergence analysis of some algorithms for solving nonsmooth equations. Math. Oper. Res. 18, 227–244 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Qi, L., Sun, D., Zhou, G.: A new look at smoothing Newton methods for nonlinear comple- mentarity problems and box constrained variational inequalities. Math. Prog. 87, 1–35 (2000)

    MathSciNet  MATH  Google Scholar 

  23. Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS PublishingCompany, Boston (1996)

    MATH  Google Scholar 

  24. Sun, D., Sun, J.: Semismooth matrix valued functions. Math. Oper. Res. 27, 150–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sun, J., Sun, D., Qi, L.: A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems, vol. 14, pp. 783–806 (2004)

  26. Toh, K.C.: Solving some large semidefinite programs via the conjugate residual method, vol. 12, pp. 669–691 (2002)

  27. Toh, K.-C., Todd, M.J., Tutuncu, R.H.: SDPT3 version 4.0 (beta) – a MATLAB software for semidefinite-quadratic-linear programming. Available: http://www.math.nus.edu.sg/mattohkc/sdpt3.html (2004)

  28. Todd, M.J.: On search directions in interior-point methods for semidefinite programming. Optim. Methods Softw. 11, 1–46 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tseng, P.: Analysis of a non-interior continuation method based on Chen-Mangasarian smoothing functions for complementarity problems. In: Fukushima, M., Qi, L. (eds.) Reformulation C Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 381–404. KluwerAcademic Publishers, Boston (1999)

  30. Tseng, P.: Merit functions for semi-definite complementarity problems. Math. Program. 83, 159–85 (1998)

    MathSciNet  MATH  Google Scholar 

  31. Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of semidefinite programming. KluwerAcademic Publishers, Boston-Dordrecht-London (2000)

    Book  MATH  Google Scholar 

  33. Yang, X., Zhang, Y., Liu, H., Pei, Y.: A Mizuno-Todd-Ye predictor-corrector infeasible-interior-point method for linear programming over symmetric cones. Numerical Algorithms. In press, available at: http://link.springer.com/article/10.1007/s11075-015-0074-7 (2015)

  34. Zhang, Y.: On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming. SIAM J. Optim. 8, 365–386 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhou, G., Toh, K.: Polynomiality of an inexact infeasible interior point algorithm for semidefinite programming, vol. 99, pp. 261–282 (2004)

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  1. School of Finance, Jiangxi University of Finance & Economics, Nanchang, 330013, China

    Aifan Ling

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  1. Aifan Ling
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Ling, A. An inexact non-interior continuation method for semidefinite programming: convergence analysis and numerical results. Numer Algor 73, 219–244 (2016). https://doi.org/10.1007/s11075-015-0093-4

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