Farkas Lemmas And Robust Approximate Solutions For Constrained Fractional Optimization Problems

In this thesis,Farkas lemmas for fractional optimization problems with DC functions(the difference of convex functions)are investigated.And the robust approximate solutions for multiobjective semi-infinite fractional optimization problems with uncertain data are also investigated.The thesis is divided into four chapters.Chapter 1,at first,the research background of fractional optimization problems is recalled.Then,the development and researches on the topic of Farkas lemmas and approximate solutions of fractional optimization problems are reviewed.Finally,the motivations and the main research work of this thesis are listed.Chapter 2 deals with some new Farkas lemmas for a class of constrained fractional optimization problems with DC functions.Following the idea due to Dinkelbach,we first associate the fractional optimization problem with a DC optimization problem.Then,by using the epigraph technique of the conjugate functions,we introduce some new regularity conditions and establish the duality relationships between the DC optimization problem and its Fenchel-Lagrange dual problem.Finally,we obtain some new Farkas lemmas for this fractional optimization problem.Furthermore,we show that the results obtained in this thesis extend and improve the corresponding results in the literature.Chapter 3 is devoted to the investigation of the robust approximate solutions for multiobjective semi-infinite fractional optimization problems with uncertain data,including robust optimality conditions,duality theories and saddle point theorems.We first obtain,by combining robust optimization and scalarization methodology,necessary and sufficient optimality conditions for robust approximate weakly efficient solutions of this optimization problem.Then,we introduce a Mixed type approximate dual problem for this optimization problem and investigate their robust approximate duality relationships.Moreover,we introduce a Lagrange function with uncertain data of this optimization problem,and obtain the corresponding robust approximate weak saddle point theorems.Chapter 4 summarizes the main results of this thesis and gives some remaining questions in the future.