Study On Some Problems In Multiobjective Optimization Problems

Multiobjective optimization problem is one of the main research fields of opti-mization theory and applications. Study on which involves many disciplines, such as: convex analysis, nonsmooth analysis, nonlinear functional analysis, and so on. And the theory and methods for the multiobjective optimization are widely used in the areas of modern economic planning, financial investment, engineering design, environmental protection, military, etc. In this thesis, we mainly study the theory of multiobjec-tive optimization in four aspects:higher order duality for multiobjective progamming problems, the optimality conditions and duality for a class of multiobjective minimax fractional programming pronlems, the optimality conditions and duality for approxi-mate solutions of multiobjective optimization problems and the scalar characterizations for approximate solutions of vector optimization problems. The main results, obtained in this dissertation, may be summarized as follows:1. In Chapter 1, firstly, we give brief introduction to the development and research sig-nificance of multiobjective optimization. And we also summarize the developments of the multiobjective optimization in five aspects associated with this thesis. Sec-ondly, we recall some basic concepts and results. Finally, we outline the contents studied in this thesis.2. Chapter 2 is committed to study the higher-order duality for nondifferentiable multiobjective programming probelms with cone constraints. First, we introduce higher-order convex functions by using a convex functional C, which can be seen as the generation of several kinds of F-convex functions. And then, we formu-late two kinds of higher-order dual models, and establish weak, stong and Huard type converse duality theorems for two kinds of higher-order dual models from the viewpoint of Fritz John and Kuhn-Tucker type necessary conditions under the generalized convexity assumptions.3. Chapter 3 studies symmetric duality for multiobjective programming problems. First, higher order symmetric duality for multiobjective fractional programming problems are presented. We formulate Mond-Weir type higher order symmetric dual model for a nondifferentiable multiobjectve fractional programming problems, and derive weak, strong and converse duality theorems under the assumptions of higher-order convexity. And then, for a pair of Mond-Weir type second order symmetric dual models, the corresponding duality theorems are established and a second order self duality theorem for the second order symmetric duals is derived. At last, certain shortcomings are pointed out in [98], and some appropriate modi-fications are given. We present a higher order cone invex function, and formulate Mond-Weir type higher order symmetric dual models and derive weak, strong and converse duality theorems from the viewpoint of Fritz John type necessary condi-tion for cone constraint optimization problems under the higher order cone invex assumptions.4. In chapter 4, we are concerned with a class of nondifferentiable multiobjective minimax fractional programming problem. First, the Kuhn-Tucker type necessary optimality condition for Geoffrion properly efficient solution is derived under the assumption of generalized Abadie constraint qualification. And then, the sufficient optimality conditions for weakly and Geoffrion properly efficient solutions are given under the generalized convexity assumptions. At last, based on the optimality conditions, we formulate Wolfe dual model and duality theorems are derived.5. In chapter 5, First, we introduce a new kind of approximate properly efficient solution for vector optimization problems and derive some properties of approxi-mate solutions. Nonlinear scalarizations via two kinds of nonlinear scalar functions are obtained for approximate solutions of vector optimization problems. And we present the linear scalarization under the cone subconvexlike assumptions. And then, we present sufficient conditions for approximate solutions of multiobjectice optimization problems by using several kinds of tangent sets and generalized di-rectional derivatives. The results are first presented in convex cases by using the cone of feasible directions, tangent cone and∈-normal set. And we consider non-convex case by employing local concepts. The sufficient conditions of approximate solutions are derived by using tangent cone, second order tangent sets and gener-alized directional derivatives. At last, we are concerned with quasi approximate weakly efficient solution of multiobjective optimization problems. Necessary and sufficient conditions are derived for the existence of quasi approximate weakly efficient solution in terms of Hadamard directional derivatives. And, we derive Kuhn-Tucker type necessary and sufficient conditions by limiting subdifferential under the generalized convexity assumptions. By introducing the notion of vector-valued Lagrangian function and∈-weakly saddle point of Lagrangian function, we formulate a Lagrange dual model, and weak and strong duality theorems for approximate solutions are derived.