The Lagrange Duality Of Fractional Optimization Problem With Composite Function

In this paper,we study the Lagrange dualities of general fractional programming prolblem and the fractional optimization problem with composite function.This paper consists five chapters.In the first chapter,the research background and the main conclusions for gen-eral fractional programming problem and the fractional optimization problem with composite function are introduced.In the second chapter,we provide some notations,conceptions and lemmas.In the third chapter,we consider the Lagrange dualities of general fractional programming problem.In the case when the involved functions are not necessarily lower semicontinuous and the sets are not necessarily closed,by using the properties of epigraph of the conjugate functions,we introduce a new constraint qualification.By using the new constraint qualification,the Lagrange duality and Farkas lemma of the fractional programming with cone constraints are established,which extend the corresponding results in the previous papers.In the fourth chapter,we study the Lagrange dualities of fractional optimization problem with composite function.By using the method of Dinkelbach,we trans-form the fractional optimization problem into the constrained optimization problem without fractional function.In the case when the functions are not necessarily lower semicontinuous and the sets are not necessarily closed,by using the epigraph and convexification technique,we introduce some new constraint qualifications.By using the new constraint qualifications,some necessary and sufficient conditions ensuring of the weak duality,the total duality and the strong duality to hold between the fractional optimization problem with composite function and its dual problem are given.In the last chapter,we summarize the content and prospect the future research.