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Smoothing Newton algorithm for the second-order cone programming with a nonmonotone line search

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Abstract

The smoothing-type algorithms, which are in general designed based on some monotone line search, have been successfully applied to solve the second-order cone programming (denoted by SOCP). In this paper, we propose a nonmonotone smoothing Newton algorithm for solving the SOCP. Under suitable assumptions, we show that the proposed algorithm is globally and locally quadratically convergent. To compare with the existing smoothing-type algorithms for the SOCP, our algorithm has the following special properties: (i) it is based on a new smoothing function of the vector-valued natural residual function; (ii) it uses a nonmonotone line search scheme which contains the usual monotone line search as a special case. Preliminary numerical results demonstrate that the smoothing-type algorithm using the nonmonotone line search is promising for solving the SOCP.

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References

  1. Alizadeh, F., Goldfarb, D.: Second-order cone optimization. Math. Program 95, 3–51 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983). Reprinted by SIAM, Philadelphia (1990)

  3. Chen, J.-S., Pan, S.H.: A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs. http://math.ntnu.edu.tw/jschen/Papers/survey.pdf. (2003)

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen, X.D., Sun, D., Sun, J.: Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems. Comput. Optim. Appl. 25, 39–56 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chi, X.N., Liu, S.Y.: A one-step smoothing Newton method for second-order cone programming. J. Comput. Appl. Math. 223, 114–123 (2009)

    Google Scholar 

  7. Chi, X.N., Liu, S.Y.: A non-interior continuation method for second-order cone optimization. Optim. 58, 965–979 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dai, Y.H.: On the nonmonotone line search. J. Optim. Theory Appl. 112, 315–330 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dai, Y.H.: A nonmonotone conjugate gradient algorithm for unconstrained optimization. J. Syst. Sci. Complex 15, 139–145 (2002)

    MATH  MathSciNet  Google Scholar 

  10. Fang, L., He, G.P., Hu, Y.H.: A new smoothing Newton-type method for second-order cone programming problems. Appl. Math. Comput. 215, 1020–1029 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fang, L., Feng, Z.Z.: A smoothing Newton-type method for second-order cone programming problems based on a new smoothing Fischer-Burmeister function. Comput. Appl. Math. 30, 569–588 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fukushima, M., Luo, Z., Tseng, P.: Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim. 12, 436–460 (2002)

    Article  MathSciNet  Google Scholar 

  13. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23, 707–716 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grippo, L., Lampariello, F., Lucidi, S.: A truncated Newton method with nonmonotone line search for unconstrained optimization. J. Optim. Theory Appl. 60, 401–419 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Grippo, L., Lampariello, F., Lucidi, S.: A class of nonmonotone stabilization method in unconstrained optimization. Numer. Math. 59, 779–805 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hu, S.L., Huang, Z.H., Wang, P.: A non-monotone smoothing Newton algorithm for solving nonlinear complementarity problems. Optim. Methods Softw. 24, 447–460 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Huang, Z.H., Han, J., Chen, Z.: A predictor-corrector smoothing Newton algorithm, based on a new smoothing function, for solving the nonlinear complementarity problem with a \(P_0\) function. J. Optim. Theory Appl. 117, 39–68 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  18. Huang, Z.H., Liu, X.H.: Extension of smoothing Newton algorithms to solve linear programming over symmetric cones. J. Syst. Sci. Complex 24, 195–206 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Huang, Z.H., Han, J.: Non-interior continuation method for solving the monotone semidefinite complementarity problem. Appl. Math. Optim. 47, 195–211 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jiang, H.: Smoothed Fischer-Burmeister equation methods for the complementarity problem. Department of Mathematics, The University of Melbourne, Parille, Victoria, Australia, June, Technical Report (1997)

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

    MATH  MathSciNet  Google Scholar 

  22. Kong, L.C., Tunçel, L., Xiu, N.H.: Equivalent conditions for Jacobian nonsingularity in linear symmetric cone programming. J. Optim. Theory Appl. 148, 364–389 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 957–972 (1977)

    Article  MathSciNet  Google Scholar 

  25. Ni, T., Wang, P.: A smoothing-type algorithm for solving nonlinear complementarity problems with a non-monotone line search. Appl. Math. Comput. 216, 2207–2214 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pataki, G., Schmieta, S.: The DIMACS library of semidefinite-quadratic- linear programs, Preliminary draft, Computational Optimization Research Center, Columbia University, New York. http://dimacs.rutgers.edu/Challenges

  27. Pan, S.H., Bi, S.J., Chen, J.S.: Nonsingular conditions for FB system of reformulating nonlinear second-order cone programming. Abstract Appl. Anal., Article ID 602735, 21 pages (2013) doi:10.1155/2013/602735

  28. Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program 58, 353–367 (1993)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  30. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (2004)

    Google Scholar 

  31. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  32. Robinson, S.M.: Local structure of feasible sets in nonlinear programming. Part III: stability and sensitivity. Math. Program Stud. 30, 45–66 (1987)

    Article  MATH  Google Scholar 

  33. Sun, D.: A regularization Newton method for solving nonlinear complementarity problems. Appl. Math. Optim. 40, 315–339 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  34. Toh K.C., Tütüncü, R.H., Todd, M.J: SDPT3 Version 3.02-A MATLAB software for semidefinite-quadratic-linear programming (2002). http://www.math.nus.edu.sg/mattohkc/sdpt3.html

  35. Tang, J.Y., He, G.P., Dong, L., Fang, L.: A smoothing Newton method for second-order cone optimization based on a new smoothing function. Appl. Math. Comput. 218, 1317–1329 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Tang, J.Y., He, G.P., Dong, L., Fang, L.: A new one-step smoothing Newton method for second-order cone programming. Appl. Math. 57, 311–331 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  37. Wang, Y., Zhang, L.W.: Nonsingularity in second-order cone programming via the smoothing metric projector. Sci. China Math. 53, 1025–1038 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. Yoshise, A.: Interior point trajectories, a homogeneous model for nonlinear complementarity problems over symmetric cones. SIAM J. Optim. 17, 1129–1153 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zhang, H.C., Hager, W.W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14, 1043–1056 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This paper was partly supported by National Natural Science Foundation of China (11101248), Excellent Young Scientist Foundation of Shandong Province (BS2011SF024, BS2012SF025), Project of Shandong Province Higher Educational Science and Technology Program (J10LA51) and Science Technology Research Projects of Education Department of Henan Province (13A110767). The authors would like to thank two referees for their valuable suggestions that greatly improved the paper. Especially, we sincerely thank Dr. Yun Wang for her discussion on the conditions equivalent to the nonsingularity of Clarke’s generalized Jacobian of the KKT nonsmooth system.

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Authors and Affiliations

  1. College of Mathematics and Information Science, Xinyang Normal University, Xinyang, 464000, China

    Jingyong Tang & Li Dong

  2. College of Mathematics and System Science, Taishan University, Tai’an, 271021, China

    Liang Fang

  3. Department of Mathematics, School of Science, Shandong University of Technology, Zibo, 255049, China

    Jinchuan Zhou

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  1. Jingyong Tang
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  2. Li Dong
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  3. Liang Fang
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  4. Jinchuan Zhou
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Correspondence to Jingyong Tang.

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Tang, J., Dong, L., Fang, L. et al. Smoothing Newton algorithm for the second-order cone programming with a nonmonotone line search. Optim Lett 8, 1753–1771 (2014). https://doi.org/10.1007/s11590-013-0699-1

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