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A New Smooth NCP Function for Solving Semidefinite Nonlinear Complementarity Problems

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Abstract

In this paper, we propose to solve semidefinite nonlinear complementarity problems (NCP) associated to a nonlinear matrix function , by a quasi-Newton method. For this, we reformulate this problem as a smooth nonlinear matrix equation by using a new smooth NCP matrix function, then we apply a quasi-Newton method for solving this matrix equation. We prove the local superlinear convergence of our algorithm and we give some numerical examples to illustrate the efficiency of the proposed method.

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References

  1. Benahmed, B., Mokhtar-Kharroubi, H., de Malafosse, B., Yassine, A.: Quasi-Newton methods in infinite-dimensional spaces and application to matrix equations. J. Glob. Optim. (2011). https://doi.org/10.1007/s10898-010-9564-2

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Sun, D., Sun, J.: Semismooth matrix-valued functions. Math. Oper. Res. 27(1), 150–169 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Sun, D., Sun, J.: Strong semismoothness of the Fischer–Burmeister SDC and SOC complementarity functions. Math. Program. 103, 575–581 (2005). https://doi.org/10.1007/s10107-005-0577-4

    Article  MathSciNet  MATH  Google Scholar 

  5. Buchholzer, H., Kanzow, C., Knabner, P., Kräutle, S.: The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions. Comput. Optim. Appl. 50, 193–221 (2011). https://doi.org/10.1007/s10589-010-9379-6

    Article  MathSciNet  MATH  Google Scholar 

  6. Han, L., Bi, S., Pan, S.: Nonsingularity of FB system and constraint nondegeneracy in semidefinite programming. Numer. Algorithms 62, 79–113 (2013). https://doi.org/10.1007/s11075-012-9567-9

    Article  MathSciNet  MATH  Google Scholar 

  7. Zhang, L., Zhang, N., Pang, L.: Differential properties of the symmetric matrix-valued Fischer–Burmeister function. J. Optim. Theory Appl. 153, 436–460 (2012). https://doi.org/10.1007/s10957-011-9962-8

    Article  MathSciNet  MATH  Google Scholar 

  8. Rui, S., Xu, C.: Inexact non-interior continuation method for monotone semidefinite complementarity problems. Optim. Lett. 6, 1411–1424 (2012). https://doi.org/10.1007/s11590-011-0337-8

    Article  MathSciNet  MATH  Google Scholar 

  9. Bi, S., Pan, S., Chen, J.-S.: Nonsingularity conditions for FB system of nonlinear SDPs. SIAM J. Optim. 21, 1392–1417 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Todd, M.J.: A study of search directions in primal-dual interior-point methods for semidefinite programming. Optim. Methods Softw. 11, 1–46 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Tseng, P.: Merit functions for semi-definite complementarity problems. Math. Program. 83, 159–185 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  12. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming. Kluwer, Boston (2000)

    Book  MATH  Google Scholar 

  13. Chen, X., Tseng, P.: Non-interior continuation methods for solving semidefinite complementarity problems. Math. Program. Ser. A 95, 431–474 (2003). https://doi.org/10.1007/s10107-002-0306-1

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, X., Qi, H., Tseng, P.: Analysis of nonsmooth symmetric-matrix-valued functions with applications to complementarity problems. SIAM J. Optim. 13(4), 960–985 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Huang, Z., Han, J.: Non-interior continuation method for solving the monotone semidefinite complementarity problem. Appl. Math. Optim. 47, 195–211 (2003). https://doi.org/10.1007/s00245-003-0765-7

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Sciences and Technology, University of Ibn Khaldoun, Tiaret, Algeria

    Mohamed Ferhaoui

  2. Department of Preparatory Training in Sciences and Technology, National Polytechnic School of Oran, Maurice Audin, Oran, Algeria

    Boubakeur Benahmed

  3. Higher School of Applied Sciences, Tlemcen, Algeria

    Lotfi Mouhadjer

Authors
  1. Mohamed Ferhaoui
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  2. Boubakeur Benahmed
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  3. Lotfi Mouhadjer
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Correspondence to Mohamed Ferhaoui.

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Communicated by Behnam Hashemi.

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Ferhaoui, M., Benahmed, B. & Mouhadjer, L. A New Smooth NCP Function for Solving Semidefinite Nonlinear Complementarity Problems. Bull. Iran. Math. Soc. 48, 3909–3936 (2022). https://doi.org/10.1007/s41980-022-00725-9

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  • DOI: https://doi.org/10.1007/s41980-022-00725-9

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