A quadratically convergent O( $$\sqrt n $$ L)-iteration algorithm for linear programming

AbstractRecently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Ye's predictor—corrector interior-point algorithm for linear programming maintains the O( $$\sqrt n $$ L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.

[1]  R. Tapia.  Role of slack variables in quasi-Newton methods for constrained optimization , 1979 .

[2]  Yinyu Ye,et al.  On the finite convergence of interior-point algorithms for linear programming , 1992, Math. Program..

[3]  F. Potra,et al.  An Interior-point Method with Polynomial Complexity and Superlinear Convergence for Linear Complementarity Problems , 1991 .

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

[5]  Renato D. C. Monteiro,et al.  Limiting behavior of the affine scaling continuous trajectories for linear programming problems , 1991, Math. Program..

[6]  Nimrod Megiddo,et al.  Homotopy Continuation Methods for Nonlinear Complementarity Problems , 1991, Math. Oper. Res..

[7]  Yin Zhang,et al.  A Study of Indicators for Identifying Zero Variables in Interior-Point Methods , 1994, SIAM Rev..

[8]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

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

[10]  Yin Zhang,et al.  On the Superlinear Convergence of Interior-Point Algorithms for a General Class of Problems , 1993, SIAM J. Optim..

[11]  Yin Zhang,et al.  On the Superlinear and Quadratic Convergence of Primal-Dual Interior Point Linear Programming Algorithms , 1992, SIAM J. Optim..

[12]  P. Boggs,et al.  On the convergence behavior of trajectories for linear programming , 1988 .

[13]  R. A. Tapia,et al.  A Superlinearly Convergent O(square root of nL)-Iteration Algorithm for Linear Programming , 1991 .

[14]  Yinyu Ye,et al.  Convergence behavior of interior-point algorithms , 1993, Math. Program..

[15]  Shinji Mizuno,et al.  Large-Step Interior Point Algorithms for Linear Complementarity Problems , 1993, SIAM J. Optim..

[16]  Zhi-Quan Luo,et al.  A Genuine Quadratically Convergent Polynomial Interior Point Algorithm for Linear Programming , 1994 .

[17]  Josef Stoer,et al.  On the complexity of following the central path of linear programs by linear extrapolation II , 1991, Math. Program..