Independence Of Interval Linear System And Interval Convex Quadratic Programming

Linear optimization and quadratic optimization have been a hot research topic in the operational research field.The common point of the two is that the constrained domain can be described as a linear system.Linear systems has always been an important research object of numerical algebra.Due to the measurement error or the instability of the variable itself,variables in mathematical models involved in actual production and life are often uncertain.Using deterministic quantities directly involved in calculation,will often get not desired results.Interval analysis is a powerful tool for dealing with these uncertainties.In this paper,we discuss the famous Farkas lemma of interval systems,the strong solvability of interval systems,the weak solvability of interval systems,and the weak solvability of interval linear quadratic programming and their dependence.The contents of this paper are as follows: Chapter one is an introduction.In the first place,we introduced the research background of interval linear systems and interval quadratic programming,Then a brief description of the definition interval quantity,interval symbolic representation and interval operation in interval theory are introduced,and finally introduces the research status and significance of interval systems with dependence and weak solvability of interval convex quadratic programming.Chapter two discussed the weak solvability of interval linear inequalities with dependence.First,interval linear inequalities and interval linear equations are introduced,the new concept of solvability of interval systems proposed By Li and others are introduced,and extended to a unified mode.Then pointed out sixteen different solvability to interval linear equations and interval linear inequalities.Similarly,the concept of solvability of interval linear inequalities consisting of equations or inequalities can be given accordingly.The sufficient and necessary conditions for the weak solvability of two different interval systems with dependence are considered,and the same idea is extended to the two existing conclusions of Hlad??k,a relatively simple new proof is given.Chapter three discussed the strong solvability of interval linear systems with dependence.First,the characteristics of(A)-strong solvability of interval linear systems with dependence are given,discussed the sufficient and necessary conditions for the(A)-strong solvability of two different interval systems.Then,we propose the condition of strong solvability of interval linear inequalities with dependence which was put forward by Hlad??k,and extend it to the dependence of two interval linear systems with equations and inequalities constraints.and derived the sufficient and necessary conditions for the existence of strong solvability.The different effects of dependence on the four interval systems are analyzed.Chapter four giving two method of checking weak optimality of the solution to two classes of interval convex quadratic programming.First,the definition of the weakly optimal solution for interval convex quadratic programming is given,discussed the interval convex quadratic programming with symbolic constraints and constrained domain are equations,and by using the concepts of the feasible direction and the KKT condition,a sufficient and necessary condition for checking whether the weakly feasible solution is a weakly optimal solution is given,and the equivalence of the two methods is verified.Then we apply the same idea to the interval convex quadratic programming with dependence in constraint domain,a sufficient and necessary conditions for checking whether the weakly feasible solution is a weakly optimal solution to interval convex quadratic programming with dependence are given.Chapter five summarizes the results of this paper,and the prospects are put forward.