I identified KKT non-uniqueness at trust-region boundary steps under under-determined interpolation and designed an optimality-aware remedy that selectively carries over linear or quadratic models. I derived update formulae and established convergence guarantees. This work provides the first systematic refinement of Conn and Toint’s underdetermined interpolation model through trust-region optimality analysis.
I proposed the ReMU family, generalising minimal-norm Hessian models; developed a truncated-Newton diagnostic and an online model-switching rule; and led the theoretical analysis of barycentric models. The method enhances robustness and efficiency in model-based optimisation, addressing the challenge of constructing reliable models from limited data for faster convergence.
I formulated least-H2 model updating for derivative-free optimisation inspired by an H2-projection, derived model coefficients, and established interpolation-error analysis. The approach regularises under-determined updates and reduces evaluation cost compared with least-Frobenius updates while preserving trust-region reliability. It provides the first Sobolev-based formulation of quadratic interpolation from an average-performance perspective.
I proposed 2D-MoSub, combining two-dimensional quadratic interpolation with trust-region steps and adaptive subspace updates. I proved poisedness/model-quality conditions and projection-error bounds leading to convergence guarantees, and demonstrated scalability to very high-dimensional settings, enabling principled randomised/subspace exploration in DFO. This work significantly enhances the scalability of derivative-free optimisation for large-scale problems.
I developed a derivative-free optimisation framework for transformed or encrypted objectives, characterised transformation classes that preserve approximate optimality, and analysed model updating and convergence under such transformations. This work extends underdetermined interpolation model-based methods to transformation settings. It provides a new perspective on noisy problems by treating noise as transformations.