Lagrange multiplier rule and scalarization of the set optimization problem in linear spaces

In this paper, we use the second set relation of six set relations pro- posed by Kuroiwa et al. [Nonlinear Analysis: Theory, Methods and Applications, 30 (1997) 1487-1496] to study Lagrange multiplier criterion and scalarization of the set optimization problem in linear spaces. Under the C-convexlikeness of the set-valued map, the existence theorem of Lagrange multiplier of the set optimiza tion problem is established. Under the C-subconvexlikeness of the set-valued map, the scalarization theorem of Lagrange multiplier of the set optimization problem is obtained. Some examples will be given to illustrate our results.