Primal–Dual–Infeasible Newton Approach for the Analytic Center Deep-Cutting Plane Method

The convergence and complexity of a primal–dual column generation and cutting plane algorithm from approximate analytic centers for solving convex feasibility problems defined by a deep cut separation oracle is studied. The primal–dual–infeasible Newton method is used to generate a primal–dual updating direction. The number of recentering steps is O(1) for cuts as deep as half way to the deepest cut, where the deepest cut is tangent to the primal–dual variant of Dikin's ellipsoid.

[1]  L. Khachiyan.  Polynomial algorithms in linear programming , 1980 .

[2]  Krzysztof C. Kiwiel,et al.  A note on some analytic center cutting plane methods for convex feasibility and minimization problems , 1996, Comput. Optim. Appl..

[3]  Yurii Nesterov,et al.  Complexity estimates of some cutting plane methods based on the analytic barrier , 1995, Math. Program..

[4]  John Darzentas,et al.  Problem Complexity and Method Efficiency in Optimization , 1983 .

[5]  Leonid Khachiyan,et al.  On the complexity of approximating the maximal inscribed ellipsoid for a polytope , 1993, Math. Program..

[6]  Michael J. Todd,et al.  Solving combinatorial optimization problems using Karmarkar's algorithm , 1992, Math. Program..

[7]  Michael J. Todd,et al.  Containing and shrinking ellipsoids in the path-following algorithm , 1990, Math. Program..

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

[9]  Yinyu Ye,et al.  Complexity analysis of the analytic center cutting plane method that uses multiple cuts , 1997, Math. Program..

[10]  Michael J. Todd,et al.  Improved Bounds and Containing Ellipsoids in Karmarkar's Linear Programming Algorithm , 1988, Math. Oper. Res..

[11]  G. Sonnevend.  New Algorithms in Convex Programming Based on a Notion of “Centre” (for Systems of Analytic Inequalities) and on Rational Extrapolation , 1988 .

[12]  Y. Ye.  Karmarkar's algorithm and the ellipsoid method , 1987 .

[13]  Jean-Philippe Vial,et al.  Shallow, deep and very deep cuts in the analytic center cutting plane method , 1999, Math. Program..

[14]  Pravin M. Vaidya,et al.  A cutting plane algorithm for convex programming that uses analytic centers , 1995, Math. Program..

[15]  Yinyu Ye,et al.  Complexity Analysis of an Interior Cutting Plane Method for Convex Feasibility Problems , 1996, SIAM J. Optim..

[16]  Jack Elzinga,et al.  A central cutting plane algorithm for the convex programming problem , 1975, Math. Program..

[17]  J. Goffin,et al.  Decomposition and nondifferentiable optimization with the projective algorithm , 1992 .

[18]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1996, Math. Program..

[19]  L. Rüedi,et al.  Ueber das Explosionstrauma des Ohres , 1942 .

[20]  Yinyu Ye,et al.  A Potential Reduction Algorithm Allowing Column Generation , 1992, SIAM J. Optim..

[21]  Z. Luo.  Analysis of a Cutting Plane Method That Uses Weighted Analytic Center and Multiple Cuts , 1997, SIAM J. Optim..

[22]  Jean-Philippe Vial,et al.  A Complexity Reduction for the Long-Step Path-Following Algorithm for Linear Programming , 1992, SIAM J. Optim..

[23]  Clóvis C. Gonzaga,et al.  Path-Following Methods for Linear Programming , 1992, SIAM Rev..