Study On Some Properties Of Solution Sets For Vector Optimization Problems

The stability analysis of the solution mappings for vector equilibrium problems is one of focus in vector optimization theory. Firstly, under new assumptions, we obtain the lower semicontinuity of the set of solutions for parametric set-valued strong vector equilibrium problems and parametric set-valued weak vector equilibrium problems. Then, we establish the stability of efficient points sets to vector optimization problems without the assumption of compactness. Finally, we give the concepts of approximate (C,(?))-proper efficient solution and approximate (C,(?))-proper efficient elements, and obtain scalarization theorem with set-valued maps by using co-radiant set.In the second chapter, we recall some basic concepts and properties which will be used in this thesis.In the third chapter, firstly, by a new assumption, we study the lower semicontinuity of the set of solutions for parametric set-valued strong vector equilibrium problems with-out the assumptions of C-strong mononiticity and C-convexity or C-concavity. Some examples are given to illustrate that the correctness of the consequences and results ex-tend the corresponding ones in the literature. Then, by using a scalarization method and a property involving the union of a family of lower semicontinuous set-valued mappings is lower semicontinuous, we obtain a sufficient condition for the lower semicontinuity of the solution mappings to parametric set-valued weak vector equilibrium problems in real Hausdorff topological vector spaces, where the assumption of monotonicity and any in-formation about the solution set are not necessary. The scalarization (f-efficient)solution set in the proof may be a general set in our paper, but not a singleton. Some examples are given to illustration that the correctness of our results.In the fourth chapter, By using quasi C-convex function and recession cone prop-erty, the stability of efficient points sets to vector optimization problems without the assumption of compactness is established. The lower part of the Painleve-Kuratowski convergence of the sets for efficient points of perturbed problems to the corresponding efficient sets for the vector optimization problems is obtained, where the perturbation are performed on both the objective function and the feasible set. Then examples are given to illustrate our main results.In the fifth chapter, for vector optimization problems (VP) with the objective function and constraint function are set valued maps, the concepts of approximate (C,∈)-properly efficient solutions and approximate (C,∈)-properly efficient elements are introduced, which extend∈-properly efficient solution introduced by Rong Weidong and Ma Yi, and an example is given to illustrate it. Then approximate (C,∈)-properly efficient solutions of vector optimization with set-valued maps are considered. Under the assumption of nearly (C,∈)-subconvexlikeness, we obtain the conclusions about the approximate solutions of vector optimization problems through associated scalar optimization problems:(x0, y0) is approximate (C,∈)-properly efficient element of problem (VP) if and only if it is∈σ_c(μ)-suboptimal element for the scalar problem corresponds to (VP). Especially, the necessary and sufficient conditions have the same error, which extends and improves corresponding ones in the literature.