Levitin-Polyak Well-posedness Of Generalized Mixed Variational Inequalities And Optimization Problems

Abstract In this paper, we mainly discuss the Levitin-Polyak well-posedness of generalized mixed variational inequalities and the scalarization method for Levitin-Polyak well-posedness of vectorial optimization problems.In first chapter, we research the Levitin-Polyak well-posedness of a gen-eralized mixed variational inequality. By constructing the gap function of a generalized mixed variational inequality, using the properties of gap function we show that, the Levitin-Polyak well-posedness of generalized mixed variational inequality is equivalent to the Levitin-Polyak well-poscdness of a corresponding minimization problem which is defined by gap function. We also discuss the suf-ficient and necessary conditions of the generalized mixed variational inequality and its Furi-Vignoli type measure properties.In second chapter, we research the Levitin-Polyak well-posedness of a gener-alized vector mixed variational inequality. Firstly, construct the Levitin-Polyak-α approximating sequences and gap function of the generalized vector mixed vari-ational inequality. Then, we proof that the problem of generalized vector mixed variational inequality and its corresponding optimization problem defined by gap function have the same solution, accordingly, derive the equivalence between the Levitin-Polyak well-posedness of generalized vector variational inequality and that of corresponding optimization problem defined by gap function. Finally, we obtain the sufficient and necessary conditions of the generalized vector variational inequality as well as its Furi-Vignoli type measure properties. In third chapter, we investigate scalarization the method for Levitin-Polyak well-posedncss of vectorial optimization problems, by virtue of non-linear scalarization function we prove the equivalent relationships between the Levitin-Polyak well-posedness of scalar optimization problems and the original vectorial optimization problems.