Generalized Convex Mappings And Efficiency With Variable Cone Structure

With arcwise-connected set, a concept of arcwise-connected convex cone mapping isdeduced from the convex-cone mapping. Geometrical features of arcwise-connected convexcone mapping is discussed, and some concept of weak arcwise-connected cone convex mappingis attained. Under weak conditions, their relationships is discussed and proved, some sufficientand necessary conditions of arcwise-connected convex cone mapping is attained, andarcwise-connected convex cone density is also derived. Introduction of cone semi-continuousmapping concept, a conclusion that arcwise-connected strictly convex cone mapping is sufficientcondition of arcwise-connected convex cone mapping is attained.In the topological vector space, quasi-cone defined by the set-valued mapping is created,and with it the concept defined certain more general control structure generalized subconvexlikemapping than the cone-convex mapping is introduced, and the relationship between them isdiscussed, then a equivalence conclusion of controlled subconvexlike mapping is attained.Introducing a controlled continuous concept, a conclusion that controlled strictly convexmapping is a sufficient condition of controlled strictly subconvexlike mapping is attained. A classof generalized controlled cone-convex mapping is derived and some of their relations isdiscussed; And based on separation theorem for convex sets, optimization problem ofgeneralized convex-cone mapping duality, such as duality and saddle point conditions, isdiscussed. Introducing a concept of cone set-value mapping with parameter controlled structure,and some relationships between the effective points (solutions) and the optimized points(solutions) with parameters to control the structure of unconstrained optimization problems isestablished, and a necessary and sufficient conditions of cone optimized contact theorem withparameters to control the structure is attained.