Research On The Optimality Of General Interval Linear Programming And The Inverse Problem Of Interval Linear Equations

Linear Programming (LP) is an important branch of operational research, the maincontent of linear programming is solving the extreme problems of linear objective func-tion under the linear constraints, it has been widely used in military, economic, man-agement and engineering technologies, helping people make the best decisions about therational use of limited human and material resources, financial and other resources. Dueto the complexity and uncertainties of the linear programming parameters, we shouldchoose some uncertain methods to solve it, due to the interval number theory is a goodmethod to deal with parameters in a range of variation situation, it’s simple and practical. Put the interval number theories together with linear programming, we get the inter-val linear programming (IvLP), it refers to the objective function or constraints containsinterval numbers. Because of interval linear programming model has the characteristicsof flexibility and adaptability, it is a good method to construct more realistic mathemat-ical model, so it is a kind of uncertain optimization method which has great developingprospects.The main work of this paper is as follows:The first chapter is an introduction. First of all, we introduce the research backgroundand the significance of interval linear programming theory. then, we introduce some basictheories and notations about interval number.In the last, we introduce the research status ofthe optimal solution and optimization in interval linear programming, and inverse problemabout interval linear equations.The second chapter discusses weak optimality of the solution to the general intervallinear programming. first of all, we propose a new form of interval linear programming,namely the general interval linear programming, it is a hybrid form of interval linearconstraints, including three basic forms of interval linear programming. Then, basedon the KKT conditions of linear programming, we obtain the necessary and sufcientconditions for determining whether a vector is a weak solution to the general intervallinear programming, and solved the open question raised by Hladik. Finally, we makethe general interval right-hand linear programming as a special case and promoted someconclusions. The third chapter discusses the strong optimality of the general interval linear pro-gramming. First of all, a clear concept of strong optimality of interval linear program-ming is given. then, based on the proposed strong solvability theory of the hybrid intervalrestraints system by Hladik, and duality theory in linear programming, we give the nec-essary and sufcient condition of checking the strong optimality of general interval linearprogramming.The fourth chapter discusses the inverse problem of interval linear equations. First ofall, we defined the concept of inverse problems and gives the research results about inter-val linear equations in literature[46]. Then we discover the incompleteness in this paperafter analyzes carefully. Finally, we make some complements about inverse problem andextend the conclusions.The fifth chapter summarizes the main results of this paper, and based on the pro-posed vision, the future work is prospected.