A Fractional Bilevel Programming Problem

The thesis deals with a certain fractional bilevel programming problem (FBP).It concerns several particular different programming problems. Necessary and sufficient conditions are presented for these problems to admit an exact penalty function formulation. The organization of this paper is as follows.In section 1, first, we state plainly that the bilevel programming problem is a type of mathematical model to represent a certain bilevel decision-making problem. Then, we summarize the extensive applied fields and prospects of the bilevel programming problem. Last, we introduce stressly the known research results and the work we do on this problem.In section 2, a certain linear fractional bilevel programming problem (LFBP) is introduced and studied in which the leader's objective function is a linear fractional function and the leader's constraints are linear, and the follower's programming is a linear programming with parameters. It is proved that, under certain conditions,the existences of the optimal solution to the problem (LFBP). under a fairly week condition, some results are explored concerned with an exact penalty function formulation of the problem (LFBP). The Lagrangian duality of this problem is presented also . In the end of this section ,an illustrative example is given. A certain nonlinear fractional bilevel programming problem (NFBP) is studied in section 3. For the problem (NFBP), we study two particular types: one can be transformed into concave bilevel programming, and the other can be transformed into convex bilevel programming, we study their own properties and various necessary and sufficient conditions to admit an exact penalty function formulation . In section 4, a type of multiple-objective fractional bilevel programming problem is introduced, which only the follower's function is multiple-objective, an exact penalty function formulation for such a problem is also proposed. The work we do and a potential area for further research are presented in section 5.To date, in the study of the common linear bilevel programming prob-lem to admit an exact penalty function formulation, it is required usually that the optimal value achieves at some, vertex of a permissible set P, the set P consisted of the vectors which conformed to the leader's and the follow's constraints . For affirmming holding this property, the existing common assumptions are that P is bounded or the leader's objective function is bounded from below over P, etc. These assumptions are not the necessary and sufficient conditions for this property to hold, and also P is not the feasible solution set for the leader's programming of the problem, so such assumptions are quite strict, for the fractional bilevel programming problem, literatures concerned with the exact penalty function formulation are few. In the paper , we generalize and improve the subject investigated and the assumptions being used , while achieving similar results via the combination of the methods and techniques which are used to fractional programming and bilevel programming study.