ˮ����AV��˵

On 30 November, doctoral researcher Nguyễn Cảnh Hùng successfully defended his dissertation at VNUHCM-University of Science. His thesis, entitled ‘Existence of Solutions and Duality for Several Classes of Nonsmooth Optimisation Problems,’ was completed under the supervision of Dr Thái Doãn Chương and Assoc. Prof. Nguyễn Lê Hoàng Anh.

The dissertation conducts a deep investigation within n-dimensional Euclidean space. The methodology employed a blend of established tools from variational analysis alongside novel techniques derived from semi-algebraic geometry.

The primary aim of the work was to establish the theoretical foundation for the existence of optimal solutions and to characterise the properties of the solution set, and to construct a comprehensive system of optimal conditions for a class of optimisation and robust optimisation problems where the objective function presents as nonsmooth and non-convex.

Academically, the work offers important new contributions, particularly in tackling the challenging and practically significant area of robust optimisation. The author introduced the concept of a ‘generalised tangent manifold at infinity,’ which furnishes an effective mechanism for investigating the lower boundedness of the infimum value. Building upon this, the dissertation demonstrated the equivalence between key stability properties, including robust M-tameness, robust properness, and the robust Palais–Smale condition, thereby establishing new results concerning the existence of solutions.

Alongside the investigation into the existence of solutions, the dissertation completed the theoretical landscape by formally constructing the duality problem. The author clarified the close relationship between the original problem and the dual problem through rigorous results covering weak duality, strong duality, and inverse duality. For problems incorporating functional and set constraints, necessary and sufficient optimal conditions were also rigorously established, drawing upon the concept of generalised convexity.

The dissertation’s results were highly praised by the Committee for both their urgency and potential applicability. The successful establishment of optimal conditions and the detailed analysis of solution existence are fundamental to the modelling and resolution of complex real-world problems. Furthermore, the innovative methods developed within the dissertation open up promising avenues for future research in related fields, such as asymptotic analysis, multi-objective optimisation, and the development of new sub-differential variations.