Abstract
The existence of equivalent scalar problems for properly efficient point of a given multiobjective optimization problem over arbitrary cones is studied by so many authors. This paper emphasizes two scalarizations, i.e., linear scalarization and conic scalarization, and studies geometrical viewpoint on the relationship between proper efficiency and these scalarizations. We also show that conic scalarization is a generalization of linear scalarization based on augmented dual cone which provides a new type of trade-off for properly efficient solutions.
Highlights
Multiobjective optimization is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously
We show that conic scalarization is a generalization of linear scalarization based on augmented dual cone which provides a new type of trade-off for properly efficient solutions
Using Hahn–Banach separation theorem and conic separation theorem, we show that conic scalarization is a generalization of linear scalarization which uses Bishop–Phelps cone for separating and scalarizing
Summary
Multiobjective optimization is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. A multiobjective optimization problem is as follows: min y; y2Y &Rp ð1Þ where Y & Rp is the feasible set in objective space. Later Geoffrion defined proper efficiency by eliminating unbounded trade-offs between objectives and studied their relation to Kuhn and Tucker’s proper efficiency and linear scalarization [6] This concept was generalized by Borwein [2, 3] and Benson [1] to problems that the objective space is ordered by closed convex cones. K and C and the following inequalities are satisfied: hyÃ; di\0 hyÃ; yi; forall y 2 K and d 2 Cnf0Rn g: In the rest of this section, the relationship between linear scalarization and properly efficient points will be mentioned. OTfhupsr,opyyÃjÃiercaenffibceieancsuybisftiytÃuties for M in Geoffrion’s a positive weighted definition vector
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