Research On Set-valued Optimization Problems In Real Linear Spaces

The efficiency of set-valued optimization problems is an important research subjectin nonlinear analysis theory. It is widely applied in variations, mathematicalprogramming, mathematical economics and control theory. Super efficiency and strongefficiency have better scalarization properties than other efficiencies. The space of theset-valued optimization problems plays a key role in the study of them. On the otherhand, convexity plays an important role in the optimization theory, hence each ofgeneralizations of convexity are concerned about. In this paper, the concepts of superefficiency and strong efficiency are introduced in real linear space which has only thelinear structure and does not have the topological structure. Under the assumption ofic-cone-convexlikeness, scalarization, the optimality conditions without derivatives andsaddle-point conclusions are obtained of super efficiency. Under the assumption ofnearly cone-subconvexlikeness, scalarization, the optimality conditions withoutderivatives and saddle-point conclusions are obtained on strong efficiency. The majorwork of the paper is as follows:The properties of the real linear space which has only the linear structure and doesnot have the topological structure are discussed. The order-bounded basis functional isdefined in real linear space. The linear functional bi-ordered decomposition theoremsand the properties of the basis functional are presented in real linear space.The concepts of nearly cone-subconvexlikeness mapping andic-cone-convexlikeness mapping are generalized to the linear space which has only thelinear structure and does not have the topological structure. The difference betweenthem with other convexties mapping are discussed in the paper. One equivalentcharacterization for ic-cone-convexlikeness mapping is obtained. By applying theequivalent characterization for ic-cone-convexlikeness mapping, an important propertyof the ic-cone-convexlikeness mapping is given.The concepts of super efficient point and super efficient element of solution setsfor set-valued optimization problems are defined in real linear space which has only thelinear structure and does not have the topological structure. By applying separationtheorem for convex sets and the definition of super efficient point, two scalarizationtheorems are obtained on super efficiency. Super efficient point is compared with efficient point and weak efficient point in the real paper. Under the assumption ofic-cone-convexlikeness, by applying the relation super efficiency with efficiency,scalarization theorem, Kuhn-Tucker and new saddle point necessary conditions areobtained respectively for set-valued optimization problem to attain its super efficientelement.The concepts of strong efficient point and strong efficient element of solution setsfor set-valued optimization problems are defined in real linear space which has only thelinear structure and does not have the topological structure. By the properties of thebasis functional, the scalarization theorems are obtained for strong efficient point.Under the assumption of nearly cone-subconvexlikeness, by applying linear functionalbi-ordered decomposition theorems in real linear space and separation theorem forconvex sets, scalarization theorems, theorem of Kuhn-Tucker saddle points,Kuhn-Tucker optimality conditionand Lagrange optimality condition,are obtained forset-valued optimization problem to attain its strong efficient element in rear linearspaces.All the paper is summed. Put forward some problems which need to be studyfurther.