Well-Posedness Of Vector Optimization Problems

In this thesis, the well-posedness of set optimization problems, symmetric vector quasi-equilibrium problems and generalized vector quasi-equilibrium problems, and Hadamard well-posedness of vector optimization problems are studied. Moreover, the well-posedness and stability properties of convex vector optimization problems and set-valued optimization problems are disscussed, respectively. This thesis is divided into eight chapters. It is organized as follows.In Chapter 1, the development and researches on the topic of wellposedness of scalar-valued problems, vector-valued problems, set-valued problems, variational inequalities and vector equilibrium problems are described. The study on the stability of vector optimization and the relationship between the stability and well-posedness are discussed. Also, the motivation is given and main works are listed.In Chapter 2, some definitions, which will be frequently used, are shown.In Chapter 3, three kinds of well-posedness for set optimization are first introduced. By virtue of a generalized Gerstewitz's function, the equivalent relations between the three kinds of well-posedness and the well-posedness of three kinds of scalar optimization problems are established, respectively. Then, sufficient and necessary conditions of well-posedness for set optimization problems are obtained by using a generalized forcing function. Finally, various criteria and characterizations of well-posedness are given for set optimization problems.In Chapter 4, a generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems is defined. By verifying the result of Hadamard well-posedness of set-valued fixed point problems, sufficient conditions for the generalized Levitin-Polyak well-posedness of symmetric vector quasi-equilibrium problems are given.In Chapter 5, the definitions of Levitin-Polyak well-posedness for two classes of generalized vector quasi-equilibrium problems are introduced. Then, some classical criteria and characterizations of the Levitin-Polyak well-posedness are investigated. And by virtue of gap functions for the generalized vector quasi-equilibrium problems, some equivalent relations are obtained between the Levitin-Polyak well-posedness for optimization problems and the Levitin-Polyak well-posedness for the generalized vector quasi-equilibrium problems. Finally, a set-valued version of Ekeland's variational principle is derived and applied to establish a sufficient condition for Levitin-Polyak well-posedness of a class of generalized vector quasi-equilibrium problems.In Chapter 6, two kinds of Hadamard well-posedness for vector-valued optimization problems are introduced. By virtue of scalarization functions, the scalarization theorems of convergence for sequences of vector-valued functions are established. Sufficient conditions of Hadamard well-posedness for vector optimization problems are obtained by using the scalarization theorems.In Chapter 7, well-posedness and stability for convex vector-valued optimization problems and set-valued optimization problems are introduced. The relationship among Gamma-convergence, P.K.convergence, pointwise convergence and continuous convergence of sequences of vector-valued functions are investigated.In Chapter 8, the results of this thesis are summarized and some discussions are made.