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An approximate lower order penalty approach for solving second-order cone linear complementarity problems

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Abstract

Based on a class of smoothing approximations to projection function onto second-order cone, an approximate lower order penalty approach for solving second-order cone linear complementarity problems (SOCLCPs) is proposed, and four kinds of specific smoothing approximations are considered. In light of this approach, the SOCLCP is approximated by asymptotic lower order penalty equations with penalty parameter and smoothing parameter. When the penalty parameter tends to positive infinity and the smoothing parameter monotonically decreases to zero, we show that the solution sequence of the asymptotic lower order penalty equations converges to the solution of the SOCLCP at an exponential rate under a mild assumption. A corresponding algorithm is constructed and numerical results are reported to illustrate the feasibility of this approach. The performance profile of four specific smoothing approximations is presented, and the generalization of two approximations are also investigated.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsekas, D.P., Nedić, A., Ozdaglar, A.E.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    MATH  Google Scholar 

  3. Billupsa, S.C., Murtyb, K.G.: Complementarity problems. J. Comput. Appl. Math. 124, 303–318 (2000)

    Article  MathSciNet  Google Scholar 

  4. Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97–138 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, J.-S.: Two classes of merit functions for the second-order cone complementarity problem. Math. Methods Oper. Res. 64, 495–519 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, J.-S.: The convex and monotone functions associated with second-order cone. Optimization 55, 363–385 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, J.-S.: SOC Functions and Their Applications, Springer Optimization and Its Applications 143. Springer, Singapore (2019)

    Google Scholar 

  8. Chen, J.-S., Chen, X., Tseng, P.: Analysis of nonsmooth vector-valued functions associated with second-order cones. Math. Program. 101, 95–117 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, J.-S., Ko, C.-H., Liu, Y.-D., Wang, S.-P.: New smoothing functions for solving a system of equalities and inequalities. Pac. J. Optim. 12(1), 185–206 (2016)

    MathSciNet  MATH  Google Scholar 

  10. Chen, J.-S., Pan, S.-H.: A descent method for a reformulation of the second-order cone complementarity problem. J. Comput. Appl. Math. 213, 547–558 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, J.-S., Pan, S.-H.: A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs. Pac. J. Optim. 8, 33–74 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Chen, J.-S., Tseng, P.: An unconstrained smooth minimization reformulation of the second-order cone complementarity problem. Math. Program. 104, 297–327 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen, L.-J., Ma, C.-F.: A modified smoothing and regularized Newton method for monotone second-order cone complementarity problems. Comput. Math. Appl. 61, 1407–1418 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, San Diego (1992)

    MATH  Google Scholar 

  16. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I and II. Springer-Verlag, New York (2003)

    MATH  Google Scholar 

  18. Faraut, J., Korányi, A.: Analysis on Symmetric Cones. Oxford University Press, New York, Oxford Mathematical Monographs (1994)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  20. Han, S.-P., Mangasrian, O.L.: Exact penalty function in nonlinear programming. Math. Program. 17, 251–269 (1979)

    Article  MathSciNet  Google Scholar 

  21. Hao, Z., Wan, Z., Chi, X.: A power penalty method for second-order cone linear complementarity problems. Oper. Res. Lett. 43, 137–142 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hayashi, S., Yamaguchi, T., Yamashita, N., Fukushima, M.: A matrix-splitting method for symmetric affine second-order cone complementarity problems. J. Comput. Appl. Math. 175, 335–353 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hayashi, S., Yamashita, N., Fukushima, M.: A combined smoothing and regularization method for monotone second-order cone complementarity problems. SIAM J. Optim. 15, 593–615 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang, Z.-H., Ni, T.: Smoothing algorithms for complementarity problems over symmetric cones. Comput. Optim. Appl. 45, 557–579 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kanzow, C., Ferenczi, I., Fukushima, M.: On the local convergence of semismooth Newton methods for linear and nonlinear second-order cone programs without strict complementarity. SIAM J. Optim. 20, 297–320 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lemke, C.E.: Bimatrix Equilibrium Points and Mathematical Programming. Manage. Sci. 11(7), 681–689 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lemke, C.E., Howson, J.T., Jr.: Equilibrium points of bimatrix games. J. Soc. Ind. Appl. Math. 12(2), 413–423 (1964)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Luo, Z.-Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  30. Mangasarian, O.L., Pang, J.S.: The extended linear complementarity problem. SIAM J. Matrix Anal. Appl. 16, 359–368 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Miao, X.-H., Qi, N., Saheya, B., Chen, J.-S.: Applying a type of SOC-functions to solve a system of equalities and inequalities under the order induced by second-order cone. Pac. J. Optim. 15(1), 1–22 (2019)

    MathSciNet  MATH  Google Scholar 

  32. Monteiro, R.D.C., Tsuchiya, T.: Polynomial convergence of primal algorithms for the second-order cone programs based on the MZ-family of directions. Math. Program. 88, 61–83 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pan, S.-H., Chen, J.-S.: A damped Gauss-Newton method for the second-order cone complementarity problem. Appl. Math. Optim. 59, 293–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Di Pillo, G., Grippo, L.: An exact penalty function method with global convergence properties for nonlinear programming problems. Math. Program. 36, 1–18 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pinar, M.C., Zenios, S.A.: On smoothing exact penalty functions for convex constrained optimization. SIAM J. Optim. 4, 486–511 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, S., Yang, X.-Q.: A power penalty method for linear complementarity problems. Oper. Res. Lett. 36, 211–214 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wu, Z.-Y., Bai, F.-S., Yang, X.-Q., Zhang, L.-S.: An exact lower order penalty function and its smoothing in nonlinear programming. Optimization 53, 51–68 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xu, X.-S., Meng, Z.-Q., Sun, J.-W., Shen, R.: A penalty function method based on smoothing lower order penalty function. J. Comput. Appl. Math. 235, 4047–4058 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Yang, X.-Q., Huang, X.-X.: Lower-order penalty methods for mathematical programs with complementarity constraints. Optim. Methods Softw. 19(6), 693–720 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Zangwill, W.I.: Nonlinear programming via penalty function. Manag. Sci. 13, 334–358 (1967)

    MathSciNet  Google Scholar 

  41. Zhang, L.-H., Yang, W.-H.: An efficient matrix splitting method for the second-order cone complementarity problem. SIAM J. Optim. 24, 1178–1205 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, China

    Zijun Hao

  2. Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan

    Chieu Thanh Nguyen & Jein-Shan Chen

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  1. Zijun Hao
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  2. Chieu Thanh Nguyen
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  3. Jein-Shan Chen
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Correspondence to Jein-Shan Chen.

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The author’s work is supported by the National Natural Science Foundation of China Nos. 11661002, 11871383), the Natural Science Fund of Ningxia (No. 2020AAC03236), the First-class Disciplines Foundation of Ningxia (No. NXYLXK2017B09). J.-S. Chen: The author’s work is supported by Ministry of Science and Technology, Taiwan.

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Hao, Z., Nguyen, C.T. & Chen, JS. An approximate lower order penalty approach for solving second-order cone linear complementarity problems. J Glob Optim 83, 671–697 (2022). https://doi.org/10.1007/s10898-021-01116-w

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  • DOI: https://doi.org/10.1007/s10898-021-01116-w

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