Research On The Farkas-type Theorems Of Interval Linear Systems And The Interval Quadratic Programs

Farkas-type lemma is an important separate theorem in optimistic theory.It is always applied in linear program and nonlinear program.Moreover,it is a theory basic of many significant results,such as the K-T condition is obtained by the Farkas lemma.Recently,more and more scholars pay their attentions to the study of interval optimizations,thus,it is necessary to generalize the Farkas-type theorem to the interval linear system.In the research of interval optimization problem,the study of interval linear program is more mature,however,there is less research for interval quadratic programming problem.In this thesis,there are two research directions: the Farkas-type lemmas of interval linear system and the properties of interval quadratic program.The main work of this thesis is as follows:The first chapter is an introduction.First of all,we introduce the research backgrounds and the significances of interval linear system and interval quadratic program theory.Then,we present some basic theories and notations about interval number.In the last,we introduce the research status of the Farkas-type theorems for interval linear system and the bounds of optimal value range for interval quadratic program.The second chapter discusses the Farkas-type theorems for eight traditional interval linear systems.First of all,we introduce the notations of eight traditional interval linear systems.Secondly,based on the Farkas-type theorem of the weak feasibility for interval linear equations,we formulate Farkas-type theorems for seven remaining situations in a unified form via universal and existential quantifiers.It is worth pointing out that Farkastype theorems can be formulated in some other different but equivalent forms,finally,we discuss the equivalent relationship between our results and those of other authors.The third chapter discusses the Farkas-type theorems for weak and strong solvability of general interval linear systems.We present Farkas-type necessary and sufficient conditions for weak and strong solvability of general interval linear systems.It is worth pointing out that there are two different but equivalent forms of the Faraks-type theorems in this section.Next,we have a discuss about the relationship between the results of Section 2 and our results in this section,each particular result of Farkas-type results established separately for weak and strong solvability and feasibility of interval linear systems is a special case of our general approach.Finally,we give a concrete example to illustrate our main results of this thesis.The fourth chapter discusses the Farkas-type theorems for AE solvability of two special general interval linear systems.First of all,we introduce the notations for AE solutions and AE solvability of interval linear system.Then we present Farkas-type theorems for AE solvability of two special general interval systems,based on the results,we have a discussion about the relationship among the results of Section 2,Section 3 and our main results in this section,and pointing out the main results of Section 2 and Section 3 are some special cases of the results in this section.Next,from the main results in this section,we also propose the Farkas-type theorems for AE solvability,(A)-strong solvability and(b)-strong solvability of traditional interval linear systems.The fifth chapter discusses the new method for computing the upper bound of optimal value in interval quadratic program.First of all,we introduce some notations and properties of quadratic program and interval quadratic programming problem.Then we present a new method to compute the upper bound of the optimal values,under weaker conditions.In this method,only primal program is taken into consideration and thus the condition that the duality gap is zero is also removed.As we all known,when using dual method(Hlad??k proposed in 2011),the difficulty is that it is not an easy task to check whether there is a zero duality gap.Next,we present an easy and efficient method for checking the zero duality gap.Moreover,some relations between the primal method and the dual method are discussed in detail.Finally,we give some illustrative examples and make some remarks.The sixth chapter discusses some properties of the lower bound of optimal values in interval quadratic program.First of all,we develop some complementary slackness conditions of a quadratic program and its Dorn dual.Then,some interesting and useful characteristics of the lower bound of interval quadratic programming problem are established based on these conditions.Finally,we also report some numerical results and remarks to give an insight into the problem discussed.The seventh chapter summarizes the main results of this thesis,and based on the proposed vision,the future work is prospected.