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Bounded homotopy path approach to the solution of linear complementarity problems

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Abstract

In this article a bounded path approach is introduced to find the solution of linear complementarity problem. The interior point method for linear programming was a kind of path-following method. The difficulty of finding a strictly feasible initial point for the interior point algorithm can be replaced appropriately by combining the interior point with homotopy method. A new homotopy function as well as a new approach are proposed to trace a homotopy path to find the solution of linear complementarity problem with various matrix classes. It is shown that the homotopy path approaching to the solution is smooth and bounded. Various classes of numerical examples are illustrated to show the effectiveness of the proposed algorithm.

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References

  1. Pang, J-S.: Complementarity problems. Handbook of global optimization. Handbook of global optimization 271–338 (1995)

  2. Kostreva, M.M., Wiecek, M.M.: Linear complementarity problems and multiple objective programming. Math. Program. 60(1–3), 349–359 (1993)

    Article  MathSciNet  Google Scholar 

  3. Ferris, M.C., Pang, J.-S.: Engineering and economic applications of complementarity problems. SIAM Rev. 39(4), 669–713 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  4. Jana, R., Das, A., Dutta, A.: On hidden \({Z}\)-matrix and interior point algorithm. Opsearch 56, 1106–1116 (2019)

    Article  MathSciNet  Google Scholar 

  5. Jana, R., Dutta, A., Das, A.: More on hidden \({Z}\)-matrices and linear complementarity problem. Linear Multilinear Algebra 69(6), 1151–60 (2019)

    Article  MathSciNet  Google Scholar 

  6. Neogy, S., Das, A.: Principal pivot transforms of some classes of matrices. Linear Algebra Appl. 400, 243–252 (2005)

    Article  MathSciNet  Google Scholar 

  7. Kumar, P.S.: Intuitionistic fuzzy zero point method for solving type-2 intuitionistic fuzzy transportation problem. Int. J. Oper. Res. 37, 418–451 (2020). https://doi.org/10.1504/IJOR.2020.105446

    Article  MathSciNet  Google Scholar 

  8. Kumar, P.S.: Developing a new approach to solve solid assignment problems under intuitionistic fuzzy environment. Int. J. Fuzzy Syst. Appl. 9, 1–34 (2019). https://doi.org/10.4018/IJFSA.2020010101

    Article  Google Scholar 

  9. Kumar, P.S.: A note on ‘a new approach for solving intuitionistic fuzzy transportation problem of type-2’. Int. J. Logist. Syst. Manage. 29, 102–129 (2018). https://doi.org/10.1504/IJLSM.2018.088586

    Article  ADS  Google Scholar 

  10. Kumar, P.S.: Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set. Int. J. Syst. Assur. Eng. Manage. 11, 189–222 (2020). https://doi.org/10.1007/s13198-019-00941-3

    Article  Google Scholar 

  11. Kumar, P.S.: Intuitionistic fuzzy solid assignment problems: a software-based approach. Int. J. Syst. Assur. Eng. Manage. (2019). https://doi.org/10.1007/s13198-019-00794-w

    Article  Google Scholar 

  12. Ahmadi, E., Jasemi, M., Monplaisir, L., Nabavi, M.A., Mahmoodi, A., Jam, P.A.: New efficient hybrid candlestick technical analysis model for stock market timing on the basis of the support vector machine and heuristic algorithms of imperialist competition and genetic. Expert Syst. Appl. 94, 21–31 (2018)

    Article  Google Scholar 

  13. Mahmoodi, A., Hashemi, L., Jasemi, M., Mehraban, S., Laliberté, J., Millar, R.C.: A developed stock price forecasting model using support vector machine combined with metaheuristic algorithms. Opsearch 60(1), 59–86 (2023)

    Article  Google Scholar 

  14. Mahmoudi, A., Hashemi, L., Jasemi, M., Pope, J.: A comparison on particle swarm optimization and genetic algorithm performances in deriving the efficient frontier of stocks portfolios based on a mean-lower partial moment model. Int. J. Fin. Econ. 26(4), 5659–5665 (2021)

    Article  Google Scholar 

  15. Mehrjoo, S., Jasemi, M., Mahmoudi, A.: A new methodology for deriving the efficient frontier of stocks portfolios: An advanced risk-return model. J. AI Data Min. 2(2), 113–123 (2014)

    Google Scholar 

  16. Mahmoodi, A., Zergani, M.J., Hashemi, L., Millar, R.: Analysis of optimized response time in a new disaster management model by applying metaheuristic and exact methods. Smart Resil. Transp. 4(1), 22–42 (2022)

    Article  Google Scholar 

  17. Hashemi, L., Mahmoodi, A., Jasemi, M., Millar, R.C., Laliberté, J.: Modeling a robust multi-objective locating-routing problem with bounded delivery time using meta-heuristic algorithms. Smart Resil. Transp. 3(3), 283–303 (2021)

    Article  Google Scholar 

  18. Mahmoodi, A., Hashemi, L., Laliberté, J., Millar, R.C.: Secured multi-dimensional robust optimization model for remotely piloted aircraft system (rpas) delivery network based on the sora standard. Designs 6(3), 55 (2022)

    Article  Google Scholar 

  19. Shapley, L.S.: A note on the lemke-howson algorithm, 175–189 (2009)

  20. Eaves, B.C., Saigal, R.: Homotopies for computation of fixed points on unbounded regions. Math. Program. 3(1), 225–237 (1972)

    Article  MathSciNet  Google Scholar 

  21. Chen, L., Han, L., Zhou, L.: Computing tensor eigenvalues via homotopy methods. SIAM J. Matrix Anal. Appl. 37(1), 290–319 (2016)

    Article  MathSciNet  Google Scholar 

  22. Han, L.: A homotopy method for solving multilinear systems with \({M}\)-tensors. Appl. Math. Lett. 69, 49–54 (2017)

    Article  MathSciNet  Google Scholar 

  23. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Finding zeroes of maps: homotopy methods that are constructive with probability one. Math. Comput. 32(143), 887–899 (1978)

    Article  MathSciNet  Google Scholar 

  24. Wang, X., Jiang, X.: A homotopy method for solving the horizontal linear complementarity problem. Comput. Appl. Math. (2013). https://doi.org/10.1007/s40314-013-0039-1

    Article  Google Scholar 

  25. Zhao, X., Zhang, S., Liu, Q.: A combined homotopy interior point method for the linear complementarity problem. J. Inform. Comput. Sci. 7, 1589–1594 (2010)

    ADS  Google Scholar 

  26. Watson, L.T.: Globally convergent homotopy methods: a tutorial. Appl. Math. Comput. 31, 369–396 (1989)

    Article  MathSciNet  Google Scholar 

  27. Yu, Q., Huang, C., Wang, X.: A combined homotopy interior point method for the linear complementarity problem. Appl. Math. Comput. 179(2), 696–701 (2006)

    Article  MathSciNet  Google Scholar 

  28. Xu, J., Liu, Q., Miao, Z.: A infeasible interior point homotopy method for solving linear complementarity problem. Int. Conf. Adv. Comput. Theory Eng. (ICACTE) 1, 1–4171420 (2010). https://doi.org/10.1109/ICACTE.2010.5578985

    Article  CAS  Google Scholar 

  29. Wang, Xiuyu, Jiang, Xingwu, Liu, Qinghuai: Interior point method for solving linear complementarity problems with \({P_*}\)-matrix. Int. Conf. Comput. Mechatron. Control Electron. Eng. 1, 39–42 (2010). https://doi.org/10.1109/CMCE.2010.5609642

    Article  ADS  CAS  Google Scholar 

  30. Fan, X., Xu, T., Gao, F., et al.: Solving nonlinear complementarity problem by a smoothing homotopy method. Taiwan. J. Math. 19(1), 51–63 (2015)

    Article  MathSciNet  Google Scholar 

  31. Das, A., Jana, R., Deepmala: Finiteness of criss-cross method in complementarity problem. In: International Conference on Mathematics and Computing, pp. 170–180 (2017). Springer

  32. Neogy, S., Das, A.: On almost type classes of matrices with \({Q}\)-property. Linear Multilinear Algebra 53(4), 243–257 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author A. Dutta is thankful to the Department of Science and Technology, Govt. of India, INSPIRE Fellowship Scheme for financial support.

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Authors and Affiliations

  1. Department of Mathematics, ITER Siksha O Anusandhan, ITER College Rd, Jagmohan Nagar, Bhubaneswar, Bhubaneswar, Odisha, 751030, India

    A. Dutta

  2. SQC & OR Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata, WB, 700108, India

    A. K. Das

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Dutta, A., Das, A.K. Bounded homotopy path approach to the solution of linear complementarity problems. OPSEARCH 61, 352–372 (2024). https://doi.org/10.1007/s12597-023-00687-4

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