ncku-talk2007-03-15

NCKU Math colloquium

DATE 2007-03-15　15:10-16:00

PLACE 數學館地下室演講廳

SPEAKER Professor Tamaki Tanaka（Graduate School of Science and Technology, Niigata University, Japan）

TITLE Scalarization methods for Set-Valued Optimization

ABSTRACT In optimization theory, set-valued analysis has been very widely developed with many applications in recent years. However, there are very few papers devoted to computational procedures for calculating optimal solutions and/or optimal values for a set-valued map. Generally, when we consider a vector optimization problem, we use some kinds of scalarization methods to get an equivalent scalar problem, and then we get an optimal solution and its value for the scalar problem much easier because the target space is one dimensional space and it is a total ordering space; see.
On the other hand, an optimization problem with a set-valued objective function is very interesting from the point of view practically as well as theoretically, but there are many possibilities to chose a suitable criterion to optimize feasible sets. Mathematical methodology on the comparison between sets is not so popular for practical researches, and hence we study characterizations of set-valued maps via scalarization and develop its computational procedure with finite steps in this paper.
Georgiev and Tanaka generalized Fan's inequality for set-valued maps by using a nonlinear scalarizing function regarded as a generalization of the Tchebyshev scalarization, which is well known and one of scalarization methods overcoming some nonconvexity in vector optimization. This kind of scalarizing function inherits some types of cone-convexity and cone-continuity from the parent set-valued map. The study on this kind of scalarizing function and its applications are also found in some other papers and it is referred to as the smallest strictly monotonic function Based on these approaches, Nishizawa et al have researched inherited properties of the scalarizing function.
The aim of this talk is to investigate how set-valued maps are characterized via scalarization and how an image set of set-valued map can be scalarized by computational procedures. In this paper, we introduce two types of characteristic functions for a set-valued map, and based on them we observe four types of scalarizing functions for sets in Euclidean space to characterize the images of set-valued maps. Moreover, we construct some reasonable computational procedure for values of each function, and we show some applications by using these results.

NCKU Math colloquium

DATE	2007-03-15　15:10-16:00

PLACE	數學館地下室演講廳

SPEAKER	Professor Tamaki Tanaka（Graduate School of Science and Technology, Niigata University, Japan）

TITLE	Scalarization methods for Set-Valued Optimization

ABSTRACT	In optimization theory, set-valued analysis has been very widely developed with many applications in recent years. However, there are very few papers devoted to computational procedures for calculating optimal solutions and/or optimal values for a set-valued map. Generally, when we consider a vector optimization problem, we use some kinds of scalarization methods to get an equivalent scalar problem, and then we get an optimal solution and its value for the scalar problem much easier because the target space is one dimensional space and it is a total ordering space; see. On the other hand, an optimization problem with a set-valued objective function is very interesting from the point of view practically as well as theoretically, but there are many possibilities to chose a suitable criterion to optimize feasible sets. Mathematical methodology on the comparison between sets is not so popular for practical researches, and hence we study characterizations of set-valued maps via scalarization and develop its computational procedure with finite steps in this paper. Georgiev and Tanaka generalized Fan's inequality for set-valued maps by using a nonlinear scalarizing function regarded as a generalization of the Tchebyshev scalarization, which is well known and one of scalarization methods overcoming some nonconvexity in vector optimization. This kind of scalarizing function inherits some types of cone-convexity and cone-continuity from the parent set-valued map. The study on this kind of scalarizing function and its applications are also found in some other papers and it is referred to as the smallest strictly monotonic function Based on these approaches, Nishizawa et al have researched inherited properties of the scalarizing function. The aim of this talk is to investigate how set-valued maps are characterized via scalarization and how an image set of set-valued map can be scalarized by computational procedures. In this paper, we introduce two types of characteristic functions for a set-valued map, and based on them we observe four types of scalarizing functions for sets in Euclidean space to characterize the images of set-valued maps. Moreover, we construct some reasonable computational procedure for values of each function, and we show some applications by using these results.