The research focused on three main areas: (1) duality of multi-objective optimisation problems with mixed constraints using higher-order tangent derivatives; (2) Lagrange duality and optimality conditions for the saddle-point of multi-objective half-infinite optimisation problems with vanishing constraints; and (3) necessary and sufficient second-order conditions for multi-objective optimisation problems.

The thesis introduced several noteworthy results, including an investigation into the properties of higher-order tangent sets, along with the development of the concept of higher-order tangent derivatives for multi-valued mappings; the study provided a generalised formulation of the dual problem, including Wolfe and Mond-Weir duality for optimisation problems with mixed constraints, while establishing optimality conditions for Benson-effective solutions and exploring the relationships between the original problem and its dual counterparts; the establishment of two Lagrange dual models, in both vector and scalar forms, for multi-objective half-infinite optimisation problems with vanishing constraints, leading to the identification of strong and weak duality relations between the original and dual problems; an examination of the relationships between strong duality and the optimality conditions for the saddle-point in specific problem models; the formulation of necessary and sufficient second-order conditions for effective solutions to multi-objective optimisation problems with vanishing constraints, through the appropriate second-order constraint qualifications.

To expand the scope of the research, future efforts will focus on investigating further properties of higher-order tangent derivatives for multi-valued mappings, as well as operations related to this extended derivative type. The study of optimality conditions and Lagrange duality for multi-objective half-infinite optimisation problems with vanishing constraints, using the graphs of conjugate functions, will also continue.

Additionally, the necessary and sufficient second-order conditions for multi-objective optimisation problems with vanishing constraints, specifically for second-order tight solutions, are expected to yield further exciting discoveries. The results of the thesis could be generalised to many other types of optimisation problems, such as non-smooth multi-objective optimisation problems with vanishing constraints or non-smooth multi-objective optimisation problems with balance constraints.

The thesis defence marked a significant milestone in doctoral researcher Trần Thiện Khải’s research career and contributed to enriching the field of Applied Mathematics in Viet Nam.

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