Set Optimization Problem And Its Lower Semicontinuity With Set-relations

Optimization problems have important applications in many aspects, such aseconomics, management science, and the dynamics. With the deepening anddevelopment of the research, the research works of vector optimization have graduallymoved towards set-valued optimization problem.For a long time, the research of set-valued optimization problems is with the basictheory of vector optimization problems. Recently, a new research method is appeared.So that, set-valued optimization problems have two types: vector optimization problemswith set-valued objective and constraint mappings and the set optimization problems,which were introduced by Kuroiwa. In the former, we formulate set-valued optimizationproblems as vector optimization problems to find a Pareto optimal solution. In the latter,we define the orderings on the sets of all images of feasible solutions under objectivemaps and find the minimal elements in the set with respect to the ordering of sets. Inpractical applications, the choice of Pareto efficient point as the optimal solution issometimes not appropriate. A simple example is that the basketball term, which won thechampionship, may not have necessarily the best player in all these basketball teams. So,Kuroiwa puts forward the set optimization problem.In this paper, set optimization problems are studied. In the first part, we introducethe set optimization problem, its research present situation, and the main researchcontents in this paper. In the second part, we mention order theory, concepts of cones inthe topological vector space, and the set-relations. Various semicontinuity of set-valuedmappings are also introduced. In the third part, the optimal solutions of set optimizationproblems are defined, and conducted to further characterize. In the fourth part, westudied the lower semicontinuity of the optimal value for parametric set optimizationproblems with set-valued maps, draw the stability of set optimization. Finally, wesummarize the main content of this thesis and outlook for future research.