(H,Ω) CONJUGATE MAPS AND (H,Ω) DUALITY THEORY IN MULTIOBJECTIVE OPTIMIZATION

Abstract

This paper is devoted to develop a duality theory for the nonlinear multiobjective optimization problems which aim to find all the efficient solutions. The (H,Ω) conjugate maps of point-to-set maps are defined, and their properties and relationships are discussed. The multiobjective optimization problem called primal problem is embedded into a family of perturbed problems, and the dual problem with multiobjectives in a wide sense, called the (H,Ω) conjugate dual problem is defined with the help of its (H,Ω) conjugate maps. The theorems, such as weak, strong and inverse (H,Ω) duality, which describe the relationships between the primal and dual problems are developed by means of the (H,Ω)-stability. The concepts of (H,Ω)-Lagrangian map and saddle-point are provided, and it is shown that the solution of the primal and the corresponding solution of the dual provide a saddle-point of the (H,Ω)-Lagrangian map. Finally, several special cases for H and Ω are discussed. Key words: conjugate map, subgradient, multiobjective optimization, efficiency, dual problem, Lagrangian map, saddle point, vector optimization.