A Contraction Theory Approach to Optimization Algorithms from Acceleration Flows

Much recent interest has focused on the design of optimization algorithms from the discretization of an associated optimization flow, i.e., a system of differential equations (ODEs) whose trajectories solve an associated optimization problem. Such a design approach poses an important problem: how to find a principled methodology to design and discretize appropriate ODEs. This paper aims to provide a solution to this problem through the use of contraction theory. We first introduce general mathematical results that explain how contraction theory guarantees the stability of the implicit and explicit Euler integration methods. Then, we propose a novel system of ODEs, namely the Accelerated-Contracting-Nesterov flow, and use contraction theory to establish it is an optimization flow with exponential convergence rate, from which the linear convergence rate of its associated optimization algorithm is immediately established. Remarkably, a simple explicit Euler discretization of this flow corresponds to the Nesterov acceleration method. Finally, we present how our approach leads to performance guarantees in the design of optimization algorithms for time-varying optimization problems.