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A new line search strategy for finding separating hyperplane in projection-based methods

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Abstract

Projection based methods are a family of efficient and applicable derivative free methods for solving systems of nonlinear monotone equations. These methods, at each iteration, use a backtracking line search to generate a hyperplane which strictly separates the current approximation from the solution set of the problem. Numerical experiments indicate that choosing an appropriate line search highly affects the efficiency of projection based methods. In this paper we introduce a new line search procedure for generating the separating hyperplane. The convergence properties of the new procedure is established in a simple and short way. Numerical results show that the new line search is very effective and increases the efficiency of projection based methods.

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References

  1. Ahookhosh, M., Amini, K., Bahrami, S.: Two derivative-free projection approaches for systems of large-scale nonlinear monotone equations. Numer. Algoritm. 64, 21–42 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Buhmiler, S., Krejić, N., Lužanin, Z.: Practical quasi-Newton algorithms for singular nonlinear systems. Numer. Algoritm. 55, 481–502 (2010)

    Article  MATH  Google Scholar 

  3. Cheng, W.: A PRP type method for systems of monotone equations. Math. Comput. Model. 50, 15–20 (2009)

    Article  MATH  Google Scholar 

  4. Dennis, J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia (1983)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. La Cruz, W., Martínez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1429–1448 (2006)

    Article  MATH  Google Scholar 

  7. La Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18, 583–599 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Li, D., Fukushima, M.: A global and superlinear convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations. SIAM J. Numer. Anal. 37, 152–172 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Li, Q., Li, D.H.: A class of derivative-free methods for large-scale nonlinear monotone equations. IMA J. Numer. Anal. 31, 1625–1635 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ortega, J.M., Rheinboldt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  11. Polyak, B.T.: Introduction to Optimization. Optimization Software, Inc. Publications Division, New York (1987)

    Google Scholar 

  12. Solodov, M.V., Svaiter, B.F.: A globally convergent inexact Newton method for systems of monotone equations. In: Fukushima, M., Qi, L. (eds.) Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, pp. 355–369. Kluwer Academic Publishers, Norwell (1998)

  13. Yan, Q.R., Peng, X.Z., Li, D.H.: A globally convergent derivative-free method for solving large-scale nonlinear monotone equations. J. Comput. Appl. Math. 234, 649–657 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Yu, Z.S., Lina, J., Sun, J., Xiao, Y.H., Liu, L., Li, Z.H.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59, 2416–2423 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Zhang, L., Zhou, W.J.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196, 478–484 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhao, Y.B., Li, D.: Monotonicity of fixed point and normal mappings associated with variational inequality and its application. SIAM J. Optim. 11, 962–973 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zhou, G., Toh, K.C.: Superlinear convergence of a Newton-type algorithm for monotone equations. J. Optim. Theory Appl. 125, 205–221 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Zhou, W.J., Li, D.H.: Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 25, 89–96 (2007)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Razi University, Kermanshah, Iran

    Keyvan Amini & Ahmad Kamandi

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  1. Keyvan Amini
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  2. Ahmad Kamandi
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Correspondence to Keyvan Amini.

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Amini, K., Kamandi, A. A new line search strategy for finding separating hyperplane in projection-based methods. Numer Algor 70, 559–570 (2015). https://doi.org/10.1007/s11075-015-9961-1

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  • DOI: https://doi.org/10.1007/s11075-015-9961-1

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