Research On The Algebraic Interval Solutions Of Interval Linear Systems And The Interval Linear Programs

In recent years,the problem of interval optimization has been paid more and more attention by scholars.Therefore,the study of interval linear system solution is further researched because the characterization of various solutions of interval linear systems is the basis of interval optimization theory.It is noted that the weak solution,the strong solution,the tolerance solution and the control solution of the interval linear systems are the real vector forms,the algebraic interval solutions(optimal solution)of interval linear systems and interval linear programming problems are less.This is a very interesting and urgent subject.The two main directions of this paper are: some kinds of algebraic interval solutions of the interval linear systems and the optimal algebraic interval solution and optimal value of standard interval linear programming problems.The main work is as follows:The first chapter is an introduction.First,we explain the research backgrounds and the significance of the algebraic interval solutions for interval linear systems and interval linear programs.Second,some basic knowledge and notations about the theory of interval in this paper are presented.In the end,we summarize the research status of the algebraic interval solutions for interval linear systems and interval linear programs.The second chapter introduces the algebraic interval solution of interval linear inequality system.First of all,we introduce the two kinds of expressions for Ax(A is a real matrix),and then introduce the algebraic interval solution,the equivalence condition and an example of interval linear inequality system Ax ? b.Second,we introduce the recognizing function of the algebraic interval solution of the interval linear inequality system,and use the recognizing function to give a sufficient and necessary condition to judge whether an interval vector is an algebraic interval solution of the interval linear inequality system.Finally,a property of the recognizing function of algebraic interval solution of interval linear inequality system is given.The third chapter discusses the optimal algebraic interval solution of interval linear programming.First of all,we discuss two kinds of expressions Ax(A is an interval matrix)which is under the condition of the interval vector x is nonnegative,then introduce the nonnegative algebraic interval solutions of interval linear equation Ax = b and interval linear inequality system Ax ? b,a sufficient and necessary condition for judging whether an interval vector is a nonnegative algebraic interval solution of interval linear equation or interval linear inequality system is given.Second,we introduce the recognizing function of the nonnegative algebraic interval solution of the two kinds of interval linear systems and the method of using the recognizing function to judge whether an interval vector is nonnegative algebraic interval solution.Then we introduce the optimal algebraic interval solution and the optimal value of the standard interval linear programming problem and the model of the standard interval linear programming problem with parameters,finally,some examples are given.The fourth chapter discusses the weak algebraic interval solutions of interval linear systems.First of all,we review the concept of weak solution and weak feasible solution of interval linear systems,then some lemmas needed in theorem proving are given,finally,we introduce the concept of weak algebraic interval solution and weak feasible algebraic interval solution of interval linear systems,judge an interval vector is weak algebraic interval solution or weak feasible algebraic interval solution of interval linear systems or not,and the weak feasibility of interval linear systems.The fifth chapter summarizes the main contents of this thesis,and based on the summary,the prospect of research direction is proposed.