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A homogeneous model for monotone mixed horizontal linear complementarity problems

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Abstract

We propose a homogeneous model for the class of mixed horizontal linear complementarity problems. The proposed homogeneous model is always solvable and provides the solution of the original problem if it exists, or a certificate of infeasibility otherwise. Our formulation preserves the sparsity of the original formulation and does not reduce to the homogeneous model of the equivalent standard linear complementarity problem. We study the properties of the model and show that interior-point methods can be used efficiently for the numerical solutions of the homogeneous problem. Numerical experiments show convincingly that it is more efficient to use the proposed homogeneous model for the mixed horizontal linear complementarity problem than to use known homogeneous models for the equivalent standard linear complementarity problem.

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References

  1. Andersen, E., Ye, Y.: A computational study of the homogeneous algorithm for large-scale convex optimization. Comput. Optim. Appl. 10(3), 243–280 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Andersen, E., Ye, Y.: On a homogeneous algorithm for the monotone complementarity problem. Math. Program. 84(2), 375–399 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anitescu, M., Lesaja, G., Potra, F.A.: Equivalence between different formulations of the linear complementarity problem. Optim. Methods Softw. 7(3), 265–290 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anstreicher, K.: On interior algorithms for linear programming with no regularity assumptions. Oper. Res. Lett. 11(7), 209–212 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonnans, J.F., Gonzaga, C.C.: Convergence of interior point algorithms for the monotone linear complementarity problem. Math. Oper. Res. 21(1), 1–25 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Czyzyk, J., Mehrotra, S., Wright, S.J.: PCx user guide. Technical Report OTC 96/01. Optimization Technology Center, Argonne National Laboratory and Northwestern University (1996)

  7. Gertz, E.M., Wright, S.J.: Object-oriented software for quadratic programming. ACM Trans. Math. Softw. 29(1), 58–81 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gondzio, J.: HOPDM (version 2.12)—a fast LP solver based on a primal-dual interior point method. Eur. J. Oper. Res. 85, 221–225 (1995)

    Article  MATH  Google Scholar 

  9. Gowda, M.S.: Reducing a monotone horizontal LCP to an LCP. Appl. Math. Lett. 8(1), 97–100 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Güler, O.: Generalized linear complementarity problems. Math. Oper. Res. 20(2), 441–448 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, K.L., Mehrotra, S.: Solution of monotone complementarity and general convex programming problems using a modified potential reduction interior point method. INFORMS J. Comput. 29(1), 36–53 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karmarkar, N.K.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lin, Y., Yoshise, A.: A homogeneous model for mixed complementarity problems over symmetric cones. Vietnam J. Math. 35, 541–562 (2007)

    MathSciNet  MATH  Google Scholar 

  14. Lustig, I., Marsten, R.E., Shanno, D.F.: On implementing Mehrotra’s predictor–corrector interior-point method for linear programming. SIAM J. Optim. 2(3), 435–449 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mehrotra, S.: On the implementation of a primal-dual interior point method. SIAM J. Optim. 2(4), 575–601 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  16. Monteiro, R.D.C., Pang, J.S.: Properties of an interior-point mapping for mixed complementarity problems. Math. Oper. Res. 21(3), 629–654 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Monteiro, R.D.C., Pang, J.S.: On two interior-point mappings for nonlinear semidefinite complementarity problems. Math. Oper. Res. 23(1), 39–60 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Petra, C.G.: Homogenization of mixed horizontal LCPs and applications. Ph.D. thesis, University of Maryland, Baltimore County (2009)

  19. Sznajder, R., Gowda, M.S.: Generalizations of P0- and P-properties; extended vertical and horizontal linear complementarity problems. Linear Algebra Appl. 223–224, 695–715 (1995)

    Article  MATH  Google Scholar 

  20. Ye, Y.: On homogeneous and self-dual algorithms for LCP. Math. Program. 76(1), 211–221 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ye, Y., Todd, M.J., Mizuno, S.: An \({O}(\sqrt{n}{L})\)-iteration homogeneous and self-dual linear programming algorithm. Math. Oper. Res. 19(1), 53–67 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yoshise, A.: Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones. SIAM J. Optim. 17(4), 1129–1153 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Yoshise, A.: A homogeneous algorithm for monotone complementarity problems over symmetric cones. Front. Sci. Ser. 49, 83 (2007)

    Google Scholar 

  24. Zhang, D., Zhang, Y.: A Mehrotra-type predictor-corrector algorithm with polynomiality and \(Q\)-subquadratic convergence. Ann. Oper. Res. 62, 131–150 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhang, Y.: Solving large-scale linear programs by interior-point methods under the MATLAB environment. Technical Report TR96-01. University of Maryland Baltimore County (1996)

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Acknowledgements

The work of Cosmin Petra was done the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The work of Florian Potra was supported by the National Science Foundation under Grant No. DMS-1311923.

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Authors and Affiliations

  1. Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, CA, 94550, USA

    Cosmin G. Petra

  2. Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD, 21250, USA

    Florian A. Potra

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  1. Cosmin G. Petra
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  2. Florian A. Potra
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Correspondence to Cosmin G. Petra.

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Petra, C.G., Potra, F.A. A homogeneous model for monotone mixed horizontal linear complementarity problems. Comput Optim Appl 72, 241–267 (2019). https://doi.org/10.1007/s10589-018-0035-x

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