Construction of multiobjective optimization theory based on new definitions of supremum and infimum in the multi-dimensional extended real space

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In this paper, solutions for a multiobjective optimization problem are defined as the infimum of a set in the multi-dimensional extended real space and mathematical theory of multiobjective optimization is constructed.First, we provided new definitions of supremum and infimum of a set in the extended real space, and investigated their properties. Particularly, taking infimum is regarded as an operator and some relationships between this operator and set theoretic operations such as the union and the algebraic sum are made clear. We introduced dividing and traversing properties of sets in the extended real space and proved that the infimum of an arbitrary set has these properties. This enabled us to characterize optimal solutions for a multiobjective optimization problem, particularly in the convex case.Secondly, concepts of conjugate mappings and subgradients for set-valued mappings in the extended real space were introduced and conjugate duality theory was developed based on those concepts. A relationship between subgradients and conjugate mappings was provided and sufficient conditions for subdifferentiability of set-valued mappings were studied. A primal multiobjective optimization problem was imbedded into a family of perturbed problems and its dual problem was defined in terms of the conjugate mapping. Some duality results were established between the primal problem and the dual problem.Furthermore, the perturbation mapping was defined for a parameterized multiobjective optimization problem, and its behavior was analyzed. From a qualitative viewpoint, continuity of the perturbation mapping was considered. On the other hand, from a quantitative viewpoint, graphical derivatives of the perturbation mapping were studied.The obtained results contribute to construct a new theory of multiobjective optimization.