Research On Interval Linear Programming

The study of interval linear programming has been widely used in scientific and engineering computing,especially in various branches of mathematics,which is of great research significance.Based on the existing interval linear programming research,this thesis studies the interval optimal solution and optimal value problem of interval linear programming in general,and the specific arrangement is as follows:The first chapter mainly introduces the background and significance of interval linear programming problems,the research status at home and abroad,the preliminary knowledge and the main research content of the thesis,and the preliminary knowl-edge details the commonly used interval linear programming methods,including BWC method,ILP method,TSM method,ITSM method,IMILP method,RTSM method and IILP method,etc.,and introduces the theoretical knowledge and advantages and disadvantages of these methods.The second chapter mainly introduces the solution set of interval linear equations,including the interval hull solution,the interval enclosure solution,the interval inner enclosure solution and some commonly used interval iterative methods of interval lin-ear equations.The range variation range of interval solutions after different correction methods（precondition and residual correction）is derived and proved.The feasibility of converting it into a system of parametric interval equations and then performing double precondition is verified by example when the coefficient matrix A is non-regular.The three chapter proposes an improved new method（ITS method）to solve the problem of interval linear programming,starting from the optimal solution of the char-acteristic model,obtaining the set of subscripts of tight constraints,introducing interval number corresponding to tight constraints,obtaining a system of interval linear equa-tions and solving,if the interval hull solution of the equation system x⁷)>0,and the calculated enclosure solution holds,then the optimal solution set can be described by∩,but finding the interval enclosure solution is only a sufficient condition for the feasibility of the method.When the interval enclosure solution r⁷)<0 is holding,the interval inner enclosure solution and the interval hull solution should be calculated to continue verification,and the numerical experimental results verify the necessity of cal-culating the inner enclosure solution and the effectiveness and practicability of the new method.The four chapter proposes an improved new method（IHLP method）to solve in-terval linear programming problems,first summarizes the definition of base B-stability and related basic knowledge of interval linear programming problems,gives a detailed method for determining the interval matrix base B-stability（regularity,feasibility,op-timality）,and obtains the optimal solution set range through the basis B-stability con-dition.The transformation of Type（A）,Type（B）and Type（C）interval programming models is discussed to improve the applicability of the method.If the optimal solution set fails the test,the optimal solution set obtained by the ILP method is contracted by borrowing the idea of improving the TSM method by THSM method,and the IHLP method for the interval linear programming problem is proposed,and the algorithm for finding the optimal solution set of interval linear programming（algorithm 4）is given,some drawbacks and a feasible way of the new method are analyzed,and finally the rationality and practicability of the new method are verified by numerical examples.