Doctoral Candidate Trần Thiện Khải successfully defended his thesis on “Application of Extended Differentiability in Some Optimisation Problems” at VNUHCM-University of Science, presenting significant research on multi-objective optimisation, duality, and optimality conditions. His work, supervised by Professors Lê Thanh Tùng and Nguyễn Lê Hoàng Anh, contributes to the advancement of applied mathematics in Viet Nam.

On 19th January, in Room F.102 at the VNUHCM-University of Science, Doctoral Candidate Trần Thiện Khải, specialising in Applied Mathematics, successfully defended his doctoral thesis on the topic “Application of Extended Differentiability in Some Optimisation Problems.”

The thesis was completed under the supervision of Associate Professor Lê Thanh Tùng and Associate Professor Nguyễn Lê Hoàng Anh. The research focused on three main areas: duality of multi-objective optimisation problems with mixed constraints using higher-order tangent derivatives, Lagrange duality, and optimality conditions for the saddle-point of multi-objective half-infinite optimisation problems with vanishing constraints, as well as necessary and sufficient second-order conditions for multi-objective optimisation problems.

Research Contents and Results

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.
• A generalised formulation of the dual problem, including Wolfe duality and Mond-Weir duality, for optimisation problems with mixed constraints. This study provided optimality conditions for effective solutions in the sense of Benson and explored the relationship 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.

Future Research Directions

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 Candidate Trần Thiện Khải’s research career and contributed to enriching the field of Applied Mathematics in Viet Nam.

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