  We expose in a tutorial fashion the mechanisms which underlie the synthesis
of optimization algorithms based on dynamic integral quadratic constraints. We
reveal how these tools from robust control allow to design accelerated gradient
descent algorithms with optimal guaranteed convergence rates by solving
small-sized convex semi-definite programs. It is shown that this extends to the
design of extremum controllers, with the goal to regulate the output of a
general linear closed-loop system to the minimum of an objective function.
  Numerical experiments illustrate that we can not only recover gradient decent
and the triple momentum variant of Nesterov's accelerated first order
algorithm, but also automatically synthesize optimal algorithms even if the
gradient information is passed through non-trivial dynamics, such as
time-delays.
