Improved automated generation of tracking bounds for first-order algorithms in time-varying optimisation

In a previous work, a general framework for automatically deriving tracking bounds for first-order algorithms in time-varying optimisation was introduced. The framework models algorithms as linear parameter-varying (LPV) systems in feedback with first-order oracles, while constructing provable bounds on the tracking error using integral quadratic constraints (IQCs). Having theoretical guarantees from this framework, it remained unclear how tight these bounds are in practice and whether they offer meaningful insights. Since the previous framework yields an infinite family of bounds based on solving semidefinite programs (SDPs), this thesis extends the framework with a heuristic designed to choose bounds to tighten the resulting tracking error certificate. Moreover, the framework is tested on a range of algorithms, including multistep versions of them. Different case studies are considered to evaluate the performance of the algorithms in terms of tracking bounds generated by the framework. A systematic approach is also proposed for modelling multistep algorithms in the framework. The results show that tightened tracking certificates can be achieved using this framework.