Study On Characterizations Of Solutions In Vector Optimization Problems

In recent years, as an important research scope of optimization theory and applica-tions, vector optimization theory and methods have been developed rapidly and become one of the main research fields in optimization. Study on which involves many disci-plines, such as: convex analysis, nonsmooth analysis, nonlinear analysis, and so on. At the same time, it has been playing an important role in many fields, such as:economics and management, engineering design, transportation, environmental protection and op-timal design, etc. We mainly focus on characterizations of solutions of vector optimiza-tion problems in four aspects:unified solution concepts and some characterizations of solutions of vector optimization problems with set-valued maps, optimality conditions of nonsmooth vector optimization problems with C(T)-valued maps, regularity conditions and (weak) strong Kuhn-Tucker optimality necessary conditions for nonsmooth vector optimization problems and characterizations of solution sets for nonlinear optimization problems with generalized convexity. This thesis includes five chapters as follows:1. In chapter1, we first give some brief introductions to the background and signif-icance of vector optimization theory and applications, especially for studying on characterizations of solutions. And we also summarize the developments of study-ing on vector optimization theory and methods in five aspects associated with this thesis. Secondly, we recall some basic concepts and results which will be used in this thesis. Finally, we outline the contents studied in this thesis.2. In chapter2, we are devoted to study some unified solution concepts and some related characterizations of solutions of vector optimization problems with set-valued maps. We first give some important characterizations of improvement set. And then, we propose a class of new generalized convexity named as nearly E-subconvexlikeness via improvement sets and establish an alternative theorem under the nearly-subconvexlikeness. Secondly, based on the ideas of Benson proper efficiency and approximate Benson proper efficiency, we propose a class of unified proper efficiency named as E-Benson proper efficiency via improvement sets and with the assumption of nearly.E-subconvexlikeness, establish scalarization theo-rem, Lagrange multipliers theorem, saddle points theorem, weak duality theorem and strong duality theorem of jE-Benson proper efficiency. Thirdly, based on the ideas of weak efficiency and approximate weak efficiency, we propose a class of unified weak efficiency named as weak E-efficiency via improvement sets and with the as-sumption of nearly E-subconvexlikeness, establish scalarization theorem, Lagrange multipliers theorem, saddle points theorem, weak duality theorem and strong dual-ity theorem of weak E-efficiency. At last, we also obtain a unified stability result in terms of improvement sets for vector optimization problems.3. In chapter3, we study some optimality conditions for vector optimization problems with C(T)-valued maps by Clarke directional derivatives and Clarke subdifferentials. Firstly, under the pseudoconvexity and by Clarke directional derivatives and Clarke subdifferentials, we give a sufficient condition of efficient solution and an equivalent condition of weak efficient solution for vector optimization problems with C(T)-valued maps. Secondly, we generalize the corresponding results to the case of vector optimization problems with C(T)-valued maps and inequality constraints. At last, in finite space, we discuss a special case of sufficient condition of efficient solution and give an equivalent version of the sufficient condition by using linear cone.4. In chapter4, we study the Kuhn-Tucker necessary optimality conditions of vector optimization problems under the regularity conditions. Firstly, we propose new reg-ularity conditions by Clarke directional derivatives and discuss some relations with some other known regularity conditions. Under the new regularity condition, we obtain weak Kuhn-Tucker necessary optimality condition of efficient solution and strong Kuhn-Tucker necessary optimality condition of Geoffrion proper efficient so-lution for nonsmooth vector optimization problems. Moreover, under the assump-tion of η-pseudolinearity, we also give some necessary and sufficient conditions of efficient solution for differentiable vector optimization problems.5. In chapter5, we study characterizations of the optimal solution sets for some classes of nonsmooth optimization problems under the suitable generalized con-vexity. Firstly, under the η-pseudolinearity, we give some properties of locally Lip-schitz77-pseudolinear function and characterize the optimal solution sets of locally Lipschitz η-pseudolinear optimization problems. Secondly, we point out some dis-advantages of some known research results on characterizations of the solution sets of nonsmooth pseudoinvex optimization problems and revise them accordingly. In the end, by Lagrange multipliers method, we also give some characterizations of the optimal solution sets for nonsmooth pseudoconvex optimization problems and nonsmooth pseudoinvex optimization problems with inequality constraints.