Optimality conditions for efficient solutions of nonconvex constrained multiobjective optimization problems via image space analysis

Abstract

This paper mainly focuses on optimality conditions for efficient solutions of a nonconvex constrained multiobjective optimization problem via image space analysis. By virtue of the Gerstewitz function and the sum of the components of the objective function, a nonlinear regular weak separation function for efficient solutions, is constructed. In order to illustrate the importance of the sum of the components of the objective function in the nonlinear regular weak separation function for efficient solutions, a nonlinear regular weak separation function for weak efficient solutions, is also given. Moreover, a nonlinear strong separation function for efficient and weak efficient solutions, is introduced. Then, some theorems of the weak and strong alternative and a global necessary and sufficient optimality condition for efficient solutions are derived by means of the nonlinear regular weak and strong separation functions for efficient solutions. A saddle point sufficient optimality condition for efficient solutions in terms of a generalized Lagrangian function associated with the nonlinear regular weak separation function, is established. Finally, under some suitable assumptions, a saddle point necessary optimality condition is obtained, which further yields a Karush/Kuhn–Tucker necessary optimality condition for efficient solutions by the Clarke subdifferential.