In this talk, we will present a high-level overview of systematic methods for analyzing and designing optimization algorithms, leveraging symbolic computations and semidefinite programming. In particular, we will introduce the performance estimation approach and demonstrate how it can be used to construct performance certificates and counterexamples, potentially in compact forms (e.g., via Lyapunov functions).

The presentation will be illustrated through concrete examples, as the core principles underlying these methods are already embedded in fundamental optimization schemes.

The methodology is implemented in the PEPit! package (https://pepit.readthedocs.io/), which enables users to apply the framework without requiring direct SDP modeling. This talk is based on joint work with great collaborators, who will be acknowledged during the presentation.