  This manuscript develops a new framework to analyze and design iterative
optimization algorithms built on the notion of Integral Quadratic Constraints
(IQC) from robust control theory. IQCs provide sufficient conditions for the
stability of complicated interconnected systems, and these conditions can be
checked by semidefinite programming. We discuss how to adapt IQC theory to
study optimization algorithms, proving new inequalities about convex functions
and providing a version of IQC theory adapted for use by optimization
researchers. Using these inequalities, we derive numerical upper bounds on
convergence rates for the gradient method, the heavy-ball method, Nesterov's
accelerated method, and related variants by solving small, simple semidefinite
programming problems. We also briefly show how these techniques can be used to
search for optimization algorithms with desired performance characteristics,
establishing a new methodology for algorithm design.
