The Study For The Dual Properties Of Three Types For Optimization Problems With Conical Constraints

This paper mainly study the approximation duality properties for compos-ite optimization problems with conic constraints,the characterization of the mixed type duality properties and the optimal solution sets for DC optimiza-tion problems with conic constraints,and the zero duality gap properties for DC composite optimization problems with conic constraints.The full text is divided into six chapters.In the first chapter.the research background and the main conclusions for composite optimization problem and DC optimization problem are introduced.In the second chapter,we give some notations,conceptions and lemmas.In the third chapter,we study the approximation duality properties for composite optimization problems with conic constraints.We introduce some new constraint qualifications by using the infimal convolution of conjugate func-tions.the epigraph technique and the properties of subdifferentials.Under the new constraint qualifications,the approximation duality gap properties,the sta-ble approximation duality gap properties,the approximation strong duality,the stable approximation strong duality,the stable approximation total duality gap properties,the stable approximation total duality and approximation optimali-ty condition between composite optimization problem and its Fenchel-Lagrange dual problem are established.In the fourth chapter,we study the mixed type duality properties for DC optimization problems with conic constraints.We introduced a mixed type dual problem for DC optimization problems with conic constraints.By studying the related properties of the mixed type duality problem,the characterization of the optimal solution sets for the DC cone constrained optimization problem are established.In the fifth chapter,we consider the zero dual gap properties for DC com-posite optimization problems with the conic constraints.We introduce some new constraint qualifications by using the infimal convolution of conjugate func-tions and the properties of subdifferentials.Under the new constraint qualifi-cations,the inf-sup type zero dual gap properties,the stable inf-sup type zero dual gap properties,the min-sup type zero dual gap properties,the stable min-sup-type zero dual gap properties and the Farkas lemma between DC composite optimization problem and its Lagrange dual problem are established.In the last chapter,we summarize the content and prospect the future research.