無中文摘要

The system of absolute value equations Ax + B|x| = b, denoted by AVEs, is a non-differentiable NP-hard problem, where A,B are arbitrary given n × n real matrices and b is arbitrary given n-dimensional vector. In this paper, we study four new smoothing functions and propose a smoothing-type algorithm to solve AVEs. With the assumption that the minimal singular value of the matrix A being strictly greater than the maximal singular value of the matrix B, we prove that the algorithm is globally and locally quadratically convergent with the four smooth equations.

[1] S. L. Hu, Z. H. Huang, and J. S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, Journal of Computational and Applied Mathematics, vol. 230, pp. 69-82, 2009.
[2] S. L. Hu, Z. H. Huang, and Q. Zhang, A generalized Newton method for absolute value equations associated with second order cones, Journal of Computational and Applied Mathematics, vol. 235, pp. 1490-1501, 2011.
[3] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
[4] Z. H. Huang, Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithms, Mathematical Methods of Operation Research, vol 61, pp. 41-45, 2005.
[5] Z-H. Huang, Y. Zhang, and W. Wu, A smoothing-type algorithm for solving system of inequalities, Journal of Computational and Applied Mathematics, vol. 220, pp. 355-363, 2008.
[6] X. Jiang, and Y. Zhang, A smoothing-type algorithm for absolute value equations, Journal of Industrial and Management Optimization, vol. 9, pp. 789-798, 2013.
[7] J. S. Chen, C. H, Ko, Y. D. Liu, and S. P. Wang, New smoothing functions for solving a system of equalities and inequalities, to appear in Pacific Journal of Optimization, 2016.
[8] O. L. Mangasarian, Absolute value equation solution via concave minimization, Optimization Letters, vol. 1, pp. 3-5, 2007.
[9] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optimization Letters, vol. 3, pp. 101-108, 2009.
[10] O. L. Mangasarian, Primal-dual bilinear programming solution of the absolute value equation, Optimization Letters, vol. 6, pp. 1527-1533, 2012.
[11] O. L. Mangasarian, Absolute value equation solution via dual complementarity, Optimization Letters, vol. 7, pp. 625-630, 2013.
[12] O. L. Mangasarian and R. R. Meyer, Absolute value equation, Linear Algebra and Its Applications, vol. 419, pp. 359-367, 2006.
[13] L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, vol. 18, pp. 227-244, 1993.
[14] L. Qi, D. Sun, and G.L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems, Mathematical Programming, vol. 87, pp. 1-35, 2000.
[15] Robert G. Bartle, The Elements of Real Analysis, Wiley, Second Edition, 1976
[16] Y. Zhang and Z-H. Huang, A nonmonotone smoothing-type algorithm for solving a system of equalities and inequalities, Journal of Computational and Applied Mathematics, vol. 233, pp. 2312-2321, 2010.
[17] C. Zhang and Q. J. Wei, Global and finite onvergence of a generalized Newton method for absolute value equations, Journal of Optimization Theory and Applications, vol. 143, pp. 391-403, 2009.