Optimality Conditions Of Approximate Solutions And Approximate Dualities For Fractional Optimization Problems

Fractional optimization problem is widely used in computer vision,image processing,economic and trade balance and many other fields,so the study of fractional optimization problem has attracted the attention of industrial departments and scientific research institutions.Therefore,the study of fractional optimization problem has important research value and academic significance both in theory and in application prospect.In this paper,we study optimality conditions of approximate solutions and approximate duality results for classical fractional optimization problem and the fractional optimization problem with composite functions.This paper is divided into five chapters.In the first chapter,the research background and the main conclusions for classical fractional programming problem and the fractional optimization problem with composite functions are introduced.In the second chapter,we provide some notations,conceptions and lemmas.In the third chapter,we study optimality conditions of approximate solutions and approximate duality results for fractional optimization problem.We introduce some new constraint qualifications by using the properties of εsubdifferentials,the infimal convolution of conjugate functions and the epigraph technique.Under the new constraint qualifications,optimality conditions of approximate solutions for fractional optimization problem,the approximation duality gap properties,the stable approximation duality gap properties,the approximation strong duality and the stable approximation strong duality between fractional optimization problem and its Fenchel-Lagrange dual problem are established,which extend the corresponding results in the previous paper.In the fourth chapter,we introduce some new constraint qualifications by using the properties of ε-subdifferentials.Under those constraint qualifications,optimality conditions of approximate solutions for the fractional optimization problem with composite functions are established.Several known results in the composite optimization problem are extended and improved.In the fifth chapter,we study approximate duality results for the fractional optimization problem with composite functions.We introduce some new constraint qualifications by using the infimal convolution of conjugate functions and the epigraph technique.Under the new constraint qualifications,the approximation duality gap properties,the stable approximation duality gap properties,the approximation strong duality and the stable approximation strong duality between the fractional optimization problem with composite functions and its Fenchel-Lagrange dual problem are established.Most of results are proper extensions of the results in the composite optimization problem.