Rate of convergence of a class of numerical methods solving linear inequality systems

The problem of finding a feasible solution to a linear inequality system arises in numerous contexts. In E. González-Gutiérrez and M.I. Todorov [A relaxation method for solving systems with infinitely many linear inequalities, Optim. Lett. DOI: 10.1007/s11590-010-0244-4] an algorithm, called extended relaxation method (ERM), which solves the feasibility problem has been proposed. Later in E. González-Gutiérrez, L. Hernández Rebollar, and M.I. Todorov [Relaxation methods for solving linear inequality systems: Converging results, preprint, CMR, 990 (2011), pp. 1–9], we consider a class of algorithms, depending on a parameter and prove their convergence. In this article, for the same class of the ERMs a linear rate of convergence is obtained.

[1]  Hao Cheng,et al.  A discretization based smoothing method for solving semi-infinite variational inequalities , 2005 .

[2]  Y. Ye,et al.  On the Complexity of a Column Generation Algorithm for Convex or Quasiconvex Feasibility Problems , 1994 .

[3]  M. A. López-Cerdá,et al.  Linear Semi-Infinite Optimization , 1998 .

[4]  Antonino Maugeri,et al.  On general infinite dimensional complementarity problems , 2007, Optim. Lett..

[5]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[6]  H. Hu,et al.  A Projection Method for Solving Infinite Systems of Linear Inequalities , 1994 .

[7]  Chih-Jen Lin,et al.  Implementation of an inexact approach to solving linear semi-infinite programming problems , 1995 .

[8]  Kam-Chau Wong,et al.  Infinite inequality systems and cardinal revelations , 2005 .

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

[10]  Gabor T. Herman,et al.  Image reconstruction from projections : the fundamentals of computerized tomography , 1980 .

[11]  G. Alistair Watson,et al.  A projected lagrangian algorithm for semi-infinite programming , 1985, Math. Program..

[12]  S. Fang,et al.  An inexact approach to solving linear semi-infinite programming problems , 1994 .

[13]  Gabor T. Herman,et al.  Image Reconstruction From Projections , 1975, Real Time Imaging.

[14]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[15]  Y. Censor,et al.  Parallel Optimization: Theory, Algorithms, and Applications , 1997 .

[16]  I. J. Schoenberg,et al.  The Relaxation Method for Linear Inequalities , 1954, Canadian Journal of Mathematics.

[17]  Marco A. López,et al.  Linear semi-infinite programming theory: An updated survey , 2002, Eur. J. Oper. Res..

[18]  G. A. Watson,et al.  Numerical Experiments with Globally Convergent Methods for Semi-Infinite Programming Problems , 1983 .

[19]  Maxim I. Todorov,et al.  A relaxation method for solving systems with infinitely many linear inequalities , 2010, Optimization Letters.

[20]  Maxim I. Todorov,et al.  Relaxation methods for solving linear inequality systems: converging results , 2012 .

[21]  Y. Censor,et al.  Parallel Optimization:theory , 1997 .

[22]  María J. Cánovas,et al.  Regularity modulus of arbitrarily perturbed linear inequality systems , 2008 .