Study On Optimality Conditions And Stability Of Optimization Problems

The theory of set-valued optimization, research on which involve in such mathematical branches as set-valued analysis, convex analysis, linear and nonlinear analysis, nonsmooth analysis, topolocial vector lattice, partial ordering theory, is one of focal point problems in the vector optimization field and finds wide applications in fixed point theory, variation problems, differential inclusions, control theory and mathematical economics. Therefore, the research for them has important learning value and certain degree of difficulty.Both the optimality conditions and stability of set-valued optimization problems are important components in the theory of set-valued optimization. The optimality conditions of set-valued optimization problems is an important foundation for developing modern algorithms;The stability of optimization problems is necessary subjects in the theory of optimization, The stability of vector optimization problems has made the rich results through studying the well-posedness of problems. But the stability of set-valued optimization problems has not been established (Huang X.X. only studies the stability of unconstraint set-valued optimization problems in the sense of upper-semicontinuty). This paper is devoted to character the optimality conditions of set-valued optimization problems in the sense of various efficiency and study systematically the stability of the sets of efficient points and efficient solutions of set-valued optimization problems. The research is carried on from two aspects. One is, based on establishing the derivative and generalized gradient of set-valued maps and saddle points of set-valued optimization problems, to character the optimality of set-valued optimization problems;the other is to study the stability of efficient solutions of set-valued optimization problems in sense of various semicontinuity and the stability of efficient points of problems in sense of subdifferential. The main points of this paper is as follows:? The optimality conditions of set-valued optimization problems with derivatives are established under efficiency in normed linear space. The concept of F-quasi-convexity is introduced. When the lower direct derivatives of objectives maps and constrained maps exist, under the assumption of nearly cone-subconvexlikeness, by using properties of set of efficient points and a separation theorem for convex sets, Kuhn-Tucker necessary conditions are obtained for set-valued optimization problems in sense of efficiency. Under the assumption of F-quasi-convexity, Kuhn-Tucker sufficient condition is obtained for set-valued optimization problems in sense of efficiency;moreover, another characterization of optimality condition for efficiency is presented by using the properties of lower direct derivative of set-valued maps at weak feasible directs. On the other hand, the derivative of set-valued is given in locally convex linear space by applying the lower semidifferentiable for set-valued maps denned by Dinh T.L. in locally convex linear space. Under convexity and quasi-convexity assumption, by applying the separation theorem, Kuhn-Tucker necessary and sufficient conditions are presented for super efficiency.? The optimality conditions of strict efficiency and strong efficiency for set-valued optimization problems are presented by the strict saddle points and strong saddle points, respectively, the strict saddle points and strong saddle points of set-valued optimization problems are defined and the equivalent characterization of strict saddle points and strong saddle points are given, respectively;The optimality conditions of strict saddle points and strong saddle points are obtained by applying the separation theorem and properties of strict and strong saddle points, respectively;The Lagrange type duality problems for efficient element of set-valued optimization under strict efficiency and strong efficiency are investigated and the weak duality , strong duality, and inverse duality theorem are presented under strict efficiency and strong efficiency, then the optimality conditions of strict efficiency and strong efficiency are presented.? The optimality conditions of strict efficiency and super efficiency for set-valued optimization problems are presented by generalized gradient under strict efficiency and super efficiency, respectively. The generalized gradient of set-valued maps under strict efficiency and strong efficiency is introduced by applying contingent epiderivative of set-valued maps. Under convexity assumption, the existences of generalized gradient of set-valued maps under strict efficiency and super efficiency are proved and the optimality conditions of strict efficiency and super efficiency is obtained, respectively.? The Well-posedness and stability of set-valued optimization problems are investigated in sense of upper semicontinuious. The pointwise well-posedness of set-valued optimization problems is presented in metric vector space, an equivalent characterization is given, The pointwise well-posedness of one type set-valued optimization problems is verified, and Ekeland's vari-ational principle for set-valued maps is proved;The B-well-posedness of set-valued optimization problems is also presented, the B-well-posedness is characterized by applying asymptotically minimizing sequence, the fact that the projects of any asymptotically minimizing sequence of set-valued optimization problems being B-well-posedness on domain space converge to the set of efficient solutions of this problem is shown, and stability of parametric set-valued optimization problems is obtained under Hausdoff semicontinu-ous.? The stability of set-valued optimization problems in sense of semi-continuity and subdifferential are investigated in normed space, the sub-differential of set-valued maps is defined in sense of super efficiency and strict efficiency, under suitable conditions, the existence and properties of subdifferential are proved by applying the cone separation theorem;When domination cone or domination cone, constrainted set, and objective maps is perturbed, the stability of set-valued optimization problems is investigated in the sense of subdifferential defined under super efficiency and in sense ofsemicontinuity;the stability of set of strictly efficient points of set-valued optimization problems is investigated in the sense of subdifferential defined under strict efficiency.