Convergence behavior of interior-point algorithms

We show that most interior-point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition. These algorithms include all path-following algorithms and some potential reduction algorithms. The result also holds for the monotone complementarity problem if a strict complementarity solution exists. In general, the limit point is a solution that maximizes the number of its nonzero components among all solutions.

[1]  S. Itoh,et al.  Variational inequalities and complementarity problems , 1978 .

[2]  L. McLinden.  An analogue of Moreau's proximation theorem, with application to the nonlinear complementarity problem. , 1980 .

[3]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[4]  Earl R. Barnes,et al.  A variation on Karmarkar’s algorithm for solving linear programming problems , 1986, Math. Program..

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

[6]  James Renegar,et al.  A polynomial-time algorithm, based on Newton's method, for linear programming , 1988, Math. Program..

[7]  C. C. Gonzaga,et al.  An algorithm for solving linear programming programs in O(n3L) operations , 1988 .

[8]  C. C. Gonzaga,et al.  An Algorithm for Solving Linear Programming Problems in O(n 3 L) Operations , 1989 .

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

[10]  Clyde L. Monma,et al.  An Implementation of a Primal-Dual Interior Point Method for Linear Programming , 1989, INFORMS J. Comput..

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

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

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

[14]  Pravin M. Vaidya,et al.  An algorithm for linear programming which requires O(((m+n)n2+(m+n)1.5n)L) arithmetic operations , 1990, Math. Program..

[15]  Renato D. C. Monteiro,et al.  An Extension of Karmarkar Type Algorithm to a Class of Convex Separable Programming Problems with Global Linear Rate of Convergence , 1990, Math. Oper. Res..

[16]  Clóvis C. Gonzaga,et al.  Large Step Path-Following Methods for Linear Programming, Part I: Barrier Function Method , 1991, SIAM J. Optim..

[17]  Robert M. Freund,et al.  Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function , 1991, Math. Program..

[18]  Yinyu Ye,et al.  An O(n3L) potential reduction algorithm for linear programming , 1991, Math. Program..

[19]  Shinji Mizuno,et al.  An $$O(\sqrt n L)$$ iteration potential reduction algorithm for linear complementarity problems , 1991, Math. Program..

[20]  Clóvis C. Gonzaga,et al.  Large Step Path-Following Methods for Linear Programming, Part II: Potential Reduction Method , 1991, SIAM J. Optim..

[21]  C. C. Gonzaga,et al.  An (O√(n) L)-Iteration Large-Step Primal-Dual Affine Algorithm for Linear Programming , 1992, SIAM J. Optim..

[22]  Paul Tseng,et al.  Complexity analysis of a linear complementarity algorithm based on a Lyapunov function , 1992, Math. Program..

[23]  Jean-Philippe Vial,et al.  A polynomial method of approximate centers for linear programming , 1992, Math. Program..

[24]  Kurt M. Anstreicher,et al.  Long steps in an O(n3L) algorithm for linear programming , 1992, Math. Program..

[25]  Roy E. Marsten,et al.  The interior-point method for linear programming , 1992, IEEE Software.

[26]  Michael J. Todd.  A Low Complexity Interior-Point Algorithm for Linear Programming , 1992, SIAM J. Optim..

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

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

[29]  Osman Güler,et al.  Existence of Interior Points and Interior Paths in Nonlinear Monotone Complementarity Problems , 1993, Math. Oper. Res..

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

[31]  Shinji Mizuno,et al.  Theoretical convergence of large-step primal—dual interior point algorithms for linear programming , 1993, Math. Program..

[32]  S. Huang,et al.  Near boundary behavior of primal—dual potential reduction algorithms for linear programming , 1993, Math. Program..