Localized solutions to interval linear equations

In this paper, we propose a new type of solutions, localized solutions, to interval linear equations Ax=b. The new solutions are based on L-localized solutions and R-localized solutions. The concepts of the three solutions are given and the characteristics of the new solutions to the interval linear equations are described by means of nonlinear inequality, as is similar to the well-known Oettli-Prager inequality which is related to the mid-point and interval radius. Some necessary conditions of the three types of solutions to the equations are given.

[1]  M. Fiedler,et al.  Linear Optimization Problems with Inexact Data , 2006 .

[2]  A. Neumaier.  Interval methods for systems of equations , 1990 .

[3]  Milan Hladík,et al.  Multiparametric linear programming: Support set and optimal partition invariancy , 2010, Eur. J. Oper. Res..

[4]  Wei Li,et al.  Fault Detection in Discrete Dynamic Systems with Uncertainty Based on Interval Optimization , 2011 .

[5]  Tong Shaocheng,et al.  Interval number and fuzzy number linear programmings , 1994 .

[6]  J. Rohn.  Systems of linear interval equations , 1989 .

[7]  Milan Hladík.  Description of Symmetric and Skew-Symmetric Solution Set , 2008, SIAM J. Matrix Anal. Appl..

[8]  Sergey P. Shary,et al.  Controllable solution set to interval static systems , 1997 .

[9]  Milan Hladík,et al.  Tolerance analysis in linear systems and linear programming , 2011, Optim. Methods Softw..

[10]  Sergey P. Shary,et al.  A New Technique in Systems Analysis Under Interval Uncertainty and Ambiguity , 2002, Reliab. Comput..

[11]  A. Ben-Tal,et al.  Adjustable robust solutions of uncertain linear programs , 2004, Math. Program..

[12]  Wei Li,et al.  Numerical solution method for general interval quadratic programming , 2008, Appl. Math. Comput..

[13]  Milan Hladík.  Optimal value range in interval linear programming , 2009, Fuzzy Optim. Decis. Mak..

[14]  M. Hladík.  Optimal value bounds in nonlinear programming with interval data , 2011 .

[15]  Baoding Liu,et al.  Theory and Practice of Uncertain Programming , 2003, Studies in Fuzziness and Soft Computing.

[16]  Jiri Rohn,et al.  An algorithm for computing the hull of the solution set of interval linear equations , 2011 .

[17]  Jiri Rohn.  A general method for enclosing solutions of interval linear equations , 2012, Optim. Lett..

[18]  Jiri Rohn,et al.  An algorithm for solving the absolute value equation , 2009 .

[19]  Milan Hladík.  Additive and multiplicative tolerance in multiobjective linear programming , 2008, Oper. Res. Lett..

[20]  Sergey P. Shary,et al.  Solving the linear interval tolerance problem , 1995 .

[21]  V. Kreinovich,et al.  On the solution sets of particular classes of linear interval systems , 2003 .

[22]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .