Research On Connectedness Of The Solution Set For Set-valued Optimization Problem

Set-valued optimization problem is one of the main research fields of vectoroptimization theory and applications. It is widely used in the areas of mathematicaleconomics, stochastic programming, variational inequality, optimal control andfuzzy programming, etc. One of the important research subjects of set-valuedoptimization problem is to investigate the connectedness properties of the solutionset, as it provides a possibility moving from one solution to other solution. It is wellknown that the globally proper efficient solution is very important in the variousefficient solution of vector optimization theory, the Henig efficient solution hasmany desirable properties and requires only the ordering cone with a base, and thestrong efficient solution has good scalarization properties. Thus, in this paper, wemainly research on the connectedness of the globally proper efficient solution set,the Henig efficient solution set and the strong efficient solution set for set-valuedoptimization problem. The main content is as follows:In chapter1, we mainly introduce the research background and history aboutsome vector optimization problems, and also analyze and summarize the domestic andoverseas scholars’ research achievements about the connectedness of the solution setfor set-valued vector optimization problem.In chapter2, we mainly study the connectedness of the globally proper efficientpoint set and globally proper efficient solution set for set-valued optimization problem.In this chapter, firstly, we introduce the concepts of the globally proper efficient pointand the f efficient point. Then, we obtain the scalarization theorem of theglobally proper efficient point set. Finally, basing on the scalarization results, wepresent the theorems of the connectedness of the globally proper efficient point setand globally proper efficient solution set. The theorems are proved under thecondition that the domain is an arcwise connected and compact subset, and theobjective function is a weak semi-continuous set-valued mapping.We discusse the connectedness of the Henig efficient solution set for set-valuedoptimization problem with constraints in chapter3. Since all the research on theconnectedness of the solution set almost only involved unconstrained set-valuedoptimization problem, and the researcher rarely discuss the connectedness of thesolution set for set-valued optimization problem with constraints. So in this paper we study the connectedness of the Henig efficient solution set for set-valued optimizationproblem with constraints, and the study is finished under the the condition that thedomain is an arcwise connected and compact subset, and the objective function is acone-arcwise connected set-valued mapping.In chapter4, we study the connectedness of the strong efficient solution set forset-valued optimization problem. In this chapter, we introduce and systematicallyresearch strong efficiency in vector optimization with set-valued maps. Firstly weintroduce the concept of the strong efficient solution on the basis of the definition ofthe strong efficient point, then we obtain the theorem of the connectedness of thestrong efficient solution set under the the condition that the domain is an arcwiseconnected subset. So far, there is no literature to study the connectedness of the strongefficient solution set, so the theorem we get in this chapter extends the relative resultsabout the connectedness of the strong efficient solution set of vector optimization withset-valued maps.In chapter5, we sum up the paper and put forward some problems which need tobe study further.