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A ROBUST INTERIOR POINT METHOD FOR COMPUTING THE ANALYTIC CENTER OF AN ILL-CONDITIONED POLYTOPE WITH ERRORS

Zhouhong Wang1, Yuhong Dai2,3, Fengmin Xu4   

  1. 1 School of Science, Beijing Jiaotong University, Beijing 100044, China;
    2 LESC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190;
    3 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China;
    4 School of Economics and Finance, Xi'an Jiaotong University, Xi'an 710061, China
  • Received:2019-01-27 Revised:2019-04-11 Online:2019-12-15 Published:2019-12-15

Zhouhong Wang, Yuhong Dai, Fengmin Xu. A ROBUST INTERIOR POINT METHOD FOR COMPUTING THE ANALYTIC CENTER OF AN ILL-CONDITIONED POLYTOPE WITH ERRORS[J]. Journal of Computational Mathematics, 2019, 37(6): 843-865.

In this paper we propose an efficient and robust method for computing the analytic center of the polyhedral set P={xRn|Ax=b, x ≥ 0}, where the matrix ARm×n is ill-conditioned, and there are errors in A and b. Besides overcoming the difficulties caused by ill-conditioning of the matrix A and errors in A and b, our method can also detect the infeasibility and the unboundedness of the polyhedral set P automatically during the computation. Detailed mathematical analyses for our method are presented and the worst case complexity of the algorithm is also given. Finally some numerical results are presented to show the robustness and effectiveness of the new method.

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[1] D.S. Atkinson and P.M. Vaidya, A scaling technique for finding the weighted analytic center of a polytope, Mathematical Programming, 57(1992), 163-192.

[2] M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear Programming:Theory and Algorithms, 3rd edition, John Wiley & Sons, New Jersey, 2006.

[3] J.F. Bonnans and C.C. Gonzaga, Convergence of interior point algorithms for the monotone linear complementarity problem, Mathematics of Operations Research, 21(1996), 1-25.

[4] J.F. Bonnans and F.A. Potra, On the convergence of the iteration sequence of infeasible path following algorithms for linear complementarity problems, Mathematics of Operations Research, 22(1997), 378-407.

[5] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.

[6] Y.H. Dai, Z.H. Wang and F.M. Xu, A primal-dual method for unfolding neutron energy spectrum from multiple activation foils, Research report, AMSS, Chinese Academy of Sciences, 2018.

[7] C.C. Gonzaga, The largest step path following algorithm for monotone linear complementarity problems, Mathematical Programming, 76(1997), 309-332.

[8] C.C. Gonzaga and R.A. Tapia, On the convergence of the Mizuno-Todd-Ye algorithm to the analytic center of the solution set, SIAM Journal on Optimization, 7(1997), 47-65.

[9] M.D. González-Lima, R.A. Tapia and F.A. Potra, On effectively computing the analytic center of the solution set by primal-dual interior-point methods, SIAM Journal on Optimization, 8(1998), 1-25.

[10] M. Kojima, N. Megiddo and S. Mizuno, A primal-dual infeasible-interior-point algorithm for linear programming, Mathematical Programming, 61(1993), 263-280.

[11] M. Kojima, S. Mizuno and A. Yoshise, A primal-dual interior point algorithm for linear programming, in Progress in Mathematical Programming:Interior-Point and Related Methods, N. Megiddo (ed.), Springer, New York, 1989, 29-47.

[12] I.J. Lustig, R.E. Marsten and D.F. Shanno, On implementing Mehrotra's predictor-corrector interior-point method for linear programming, SIAM Journal on Optimization, 2(1992), 435-449.

[13] N. Megiddo, Pathways to the optimal set in linear programming, in Progress in Mathematical Programming:Interior-Point and Related Methods, N. Megiddo (ed.), Springer, New York, 1989, 131-158.

[14] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM Journal on Optimization, 2(1992), 575-601.

[15] S. Mehrotra, Quadratic convergence in a primal-dual method, Mathematics of Operations Research, 18(1993), 741-751.

[16] S. Mizuno, Polynomiality of infeasible-interior-point algorithms for linear programming, Mathematical Programming, 67(1994), 109-119.

[17] S. Mizuno, M.J. Todd and Y. Ye, On adaptive-step primal-dual interior-point algorithms for linear programming, Mathematics of Operations Research, 18(1993), 964-981.

[18] S. Mizuno, M.J. Todd and Y. Ye, A Surface of Analytic Centers and Primal-Dual InfeasibleInterior-Point Algorithms for Linear Programming, Mathematics of Operations Research, 20(1995), 135-162.

[19] R. Monteiro, J. O'Neal and T. Tsuchiya, Uniform boundedness of a preconditioned normal matrix used in interior-point methods, SIAM Journal on Optimization, 15(2004), 96-100.

[20] A. Oliveira and D. Sorensen, A new class of preconditioners for large-scale linear systems from interior point methods for linear programming, Linear Algebra and its Applications, 394(2005), 1-24.

[21] F. Potra, An infeasible-interior-point predictor-corrector algorithm for linear programming, SIAM Journal on Optimization, 6(1996), 19-32.

[22] J. Renegar, A Mathematical View of Interior-Point Methods in Convex Optimization, SIAM, Philadelphia, 2001.

[23] C. Roos, A full-newton step o(n) infeasible interior-point algorithm for linear optimization, SIAM Journal on Optimization, 16(2006), 1110-1136.

[24] C. Roos, T. Terlaky and J.P. Vial, Interior Point Methods for Linear Optimization, 2nd edition, Springer, Berlin, 2005.

[25] A. Wächter and L.T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, 106(2006), 25-57.

[26] Y. Wang, Y. Yuan and H. Zhang, A trust region-CG algorithm for deblurring problem in atmospheric image reconstruction, Science in China Series A:Mathematics, 45(2002), 731-740.

[27] M. Wright, Ill-conditioning and computational error in interior methods for nonlinear programming, SIAM Journal on Optimization, 9(1998), 84-111.

[28] S.J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, 1996.

[29] S.J. Wright, Effects of finite-precision arithmetic on interior-point methods for nonlinear programming, SIAM Journal on Optimization, 12(2001), 36-78.

[30] Y. Ye, Interior Point Algorithms:Theory and Analysis, John Wiley & Sons, New Jersey, 1997.

[31] Y. Ye, O. Güler, R.A. Tapia and Y. Zhang, A quadratically convergent O(√nL)-iteration algorithm for linear programming, Mathematical Programming, 59(1993), 151-162.

[32] Y. Ye, M.J. Todd and S. Mizuno, An O(√nL)-iteration homogeneous and self-dual linear programming algorithm, Mathematics of Operations Research, 19(1994), 53-67.

[33] Y. Zhang, On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem, SIAM Journal on Optimization, 4(1994), 208-227.

[34] Y. Zhang, Solving large-scale linear programs by interior-point methods under the MATLAB environment, Optimization Methods and Software, 10(1998), 1-31.

[35] Y. Zhangsun, Unfolding Method Based on Entropy Theory for the Determination of Neutron Spectrum (in Chinese), Master's thesis, Northwest Institute of Nuclear Technology, Xi'an, Shanxi, P. R. China, 2015.
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