Complexity of Predictor-Corrector Algorithms for LCP Based on a Large Neighborhood of the Central Path

The predictor-corrector approach for following the central path of monotone linear complementarity and linear programming problems is simple, elegant, and efficient. Although it has excellent theoretical properties when working in narrow neighborhoods of the central path, its proved complexity assumes a frustratingly high value of O(n1.5L) iterations when based on an $l_{\infty}$ neighborhood and several Newton corrector steps per iteration. This paper shows that by carefully specifying the line searches in each step, the complexity assumes the value O(nL), as should be expected for a method based on this neighborhood.

[1]  Shinji Mizuno,et al.  A new polynomial time method for a linear complementarity problem , 1992, Math. Program..

[2]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part I: Linear programming , 1989, Math. Program..

[3]  Yinyu Ye,et al.  An Asymptotical O(√(n) L)-Iteration Path-Following Linear Programming Algorithm That Uses Wide Neighborhoods , 1996, SIAM J. Optim..

[4]  Nimrod Megiddo,et al.  A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems , 1991, Lecture Notes in Computer Science.

[5]  Shinji Mizuno,et al.  On Adaptive-Step Primal-Dual Interior-Point Algorithms for Linear Programming , 1993, Math. Oper. Res..

[6]  Renato D. C. Monteiro,et al.  Interior path following primal-dual algorithms. part II: Convex quadratic programming , 1989, Math. Program..

[7]  Sanjay Mehrotra,et al.  On the Implementation of a Primal-Dual Interior Point Method , 1992, SIAM J. Optim..

[8]  B. Jansen,et al.  Primal-dual algorithms for linear programming based on the logarithmic barrier method , 1994 .

[9]  Jacek Gondzio,et al.  Multiple centrality corrections in a primal-dual method for linear programming , 1996, Comput. Optim. Appl..

[10]  Kurt M. Anstreicher,et al.  A New Infinity-Norm Path Following Algorithm for Linear Programming , 1995, SIAM J. Optim..

[11]  Richard A. Tapia,et al.  On the Convergence of the Mizuno-Todd-Ye Algorithm to the Analytic Center of the Solution Set , 1997, SIAM J. Optim..

[12]  Yin Zhang,et al.  A quadratically convergent O( $$\sqrt n $$ L)-iteration algorithm for linear programming , 1993, Math. Program..

[13]  Yin Zhang,et al.  On polynomiality of the Mehrotra-type predictor—corrector interior-point algorithms , 1995, Math. Program..

[14]  M. Kojima,et al.  A primal-dual interior point algorithm for linear programming , 1988 .

[15]  N. Megiddo Pathways to the optimal set in linear programming , 1989 .

[16]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, STOC '84.

[17]  Shinji Mizuno,et al.  A polynomial-time algorithm for a class of linear complementarity problems , 1989, Math. Program..