Semicontinuity Of Solutions For Parametric Set Optimization Problems By Scalarization Methods

Set-valued optimization problems have been studied in many fields and play a key role in operations research,which mean objective functions or constraint functions are set-valued mappings.Stability analysis occupies a central position in mathematical programming,including semicontinuity,Lipschitz continuity,H?lder continuity and so on.The semicontinuity of solution mappings for parametric set-valued optimization problems under set optimization criterion(i.e.,set optimization problems)is very meaningful and interesting.The scalarization methods are interesting and effective,which can be applied to set optimization problems.First,we define a new monotonicity of set-valued mappings by using scalarization functionals,including linear scalarization functionals and Gerstewitz functionals used to characterize set order relations.Thus we obtain sufficient conditions of the semicontinuity to the solution mappings for the parametric set optimization problems.Especially,the continuity of the set-valued objective mapping can be replaced by the monotonicity.Moreover,there does not exist any relations between the continuity and monotonicity of set-valued mappings.Then,the semicontinuity of the solution mappings for the parametric set optimization problems can be also obtained by using monotonicity assumption that defined by the oriented distance function and other suitable conditions.Therefore,the semicontinuity of solution mappings for both convex and nonconvex parametric set optimization problems can be solved when the set-valued objective mapping is not continuous,and using the oriented distance function does not require the nonempty topological interior of the ordering cone.