Convex interpolation and performance estimation of first-order methods for convex optimization

The goal of this thesis is to show how to derive in a completely automated way exact and global worst-case guarantees for first-order methods in convex optimization. To this end, we formulate a generic optimization problem looking for the worst-case scenarios. The worst-case computation problems, referred to as performance estimation problems (PEPs), are intrinsically infinite-dimensional optimization problems formulated over a given class of objective functions. To render those problems tractable, we develop (smooth and non-smooth) convex interpolation framework, which provides necessary and sufficient conditions to interpolate our objective functions. With this idea, we transform PEPs into solvable finite-dimensional semidefinite programs, from which one obtains worst-case guarantees and worst-case functions, along with the corresponding explicit proofs. PEPs already proved themselves very useful as a tool for developing convergence analyses of first-order optimization methods. Among others, PEPs allow obtaining exact guarantees for gradient methods, along with their inexact, projected, proximal, conditional, decentralized and accelerated versions.